hyperventilation.

Hematopoietic Toxicity:

and skin necrosis

Acute tubular necrosis.

Pulmonary System: Hyperventilation.

Involvement.

Table 5. Comparative analysis of clinical symptoms induced by radiation and/or Radiation Toxins – (SRD group 1-4)- induced ARS and those produced by Toxic Shock Syndrome

Hematopoietic Radiation Toxin-Thrombocytopenia, Lymphocytopenia, Granulocytopenia. Pancytopenia.

Cutaneous System Involvement: Possibleaction of any type of Radiation Toxins. Swelling, edema, blustering, desquamation, hair loss, ulceration

severe.

**Clinical symptoms induced by Bacterial**

Central Nervous System involvement

Cardio-Vascular System involvement Systolic Blood Pressure :< 90 mmHg Tachycardia >100 beats/min,

Hypotension, Hyperventilation, Loss of

Gastro-Intestinal System Involvement Vomiting, diarrhea, hematochezia and stool mucous (depending on which

Bacterial Superantigen hematotoxicity Thrombocytopenia, Platelet count < 100,000/mm3.- Thrombocytopenia., leukopenia, pancytopenia. Disseminated

Intravascular Coagulation (DIC)

Cutaneous involvement: diffuse

rash, intense erythroderma,

toxic shock syndrome).

Organ Involvement.

Pulmonary System: Hyperventilation with

maculopapular, petechial or ecchymotic

desquamation (e.g. meningococcemia or

Multiple Organ Failure and Multiple

Renal failure, Renal hypo-perfusion; Oliguria, Acute tubular necrosis

Respiratory alkalosis, pulmonary

induced by Bacterial Superantigene.

hypertension and edema, hypoxemia, ARDS

Hepatic failure and inflammation Hepatic failure

Sympathetic responsiveness .

Peripheral nervous system Confusion, Irritability, Lethargy

**Superantigene**

enteric toxin).

Syndrome.

cell receptors by cells containing major histocompatability complex class II molecules, without processing by antigen-presenting cells [32,48,86]. Superantigens (SAgs) are a class of antigens which cause non-specific activation of T-cells resulting in polyclonal T cell activation and massive cytokine release. The large number of activated T-cells secrete large amounts of cytokines (the most important of which is TNF-alpha). [32,86]. SAg stimulation of antigen presenting cells and T-cells elicits a response that is mainly inflammatory, focused on the action of TH-1 T-helper cells. Some of the major products are IL-1, IL-2, IL-6, TNF-α, gamma interferon (IFN-γ), macrophage inflammatory protein 1α (MIP-1α), MIP-1β, and also monocyte chemoattractant protein 1(MCP-1). Clinical symptoms / signs induced by bacterial Superantigenes include central and peripheral nervous system involvement (confusion, lethargy, vomiting, anorexia), involvement of the cardiovascular system with development of hypotenison, tachycardia and loss of sympathetic responsiveness. Bacterial Superantigenes activate toxic processes within the gastrointestinal system including acute vomiting and diarrhea. The important signs and clinical picture of bacterial superantigen hematotoxicity with development thrombocytopenia, leukopenia, lyphocytopenia and development of coagulopathy. Bacterial Superantigenes could activate injury of cutaneus system and diffuse rash, erythrodermia, desquamation. [32,86] Radiation Toxins of SRD groups 1-4 can activate similar pathological processes, clinical picture and clinical signs as the bacterial superantigent toxicity and the three entities: 1) Acute Radiation Syndromes, 2) SRD RT- induced acute syndromes and Bacterial Toxin- induced sepstic reaction and toxicity can be compared. (see Table 5)

#### **4.2 Comparative analysis of toxins produced by venom toxins and radiation toxins (see table 6)**

Neurotoxic proteins can be isolated from various snake venoms. These proteins can potentially be a mix of neurotoxins ,which attack the nervous system; hemototoxins, which attack the circulatory system; plus cytotoxins, bungarotoxins and many other toxins that affect the body in different ways [6,29,49,60,61,82,83,84,85]. Almost all snake venom contains hyaluronidase, an enzyme that ensures rapid diffusion of the venom into the body of the bite-victim [82,84]. Viper venoms from the Elapidae, Hydrophiidae, Atractaspididae, Viperidae and Colubridae families contain at least 25 separate classes of biologically active compounds, including enzymes and non-enzymatic molecules. Venom- neurotoxins activate different reactions with nerve synapse – presynaptic and postsynaptic action; and induce anticholinesterase-induced flaccid paralysis. Myotoxins from Venom could induce systemic skeletal muscle damage. Hemototoxins induce the development of severe microcirculatory injury, damage to vascular membranes, thrombosis or bleeding. [6, 82, 84].

#### **4.3 Radiation induced apoptosis and necrosis**

Radiation induced Apoptosis and Necrosis as response of eukaryotic cells to influence of different types of radiation remain contraversion. Radiation induced cell death by triggering apoptosis pathways was described in many articles and supported by many scientists [11, 15, 17, 39, 51, 52, 53, 54, 55, 56; 57, 58, 59, 62, 63, 74, 75, 76, 78]. However some scientists and some institutions described processes developing in irradiated eukaryotic cells as necrosis [24;34;38] and some scientists describing a variety of complex mechanism and mentioned that radiation-induced cell death could developing under different mechanisms of pathogenesis which include and apoptosis and necrosis. [1, 34, 38, 43, 77] Activation of

Radiation Toxins – Effects of Radiation Toxicity, Molecular Mechanisms of

Morphologically cells shrinks, become denser, condensation occur, original name – shrinkage necrosis. Mitochondria's structure and functions are not affected.

Karyorhexis – pyknotic nuclear fragments.

DNA broken into segments.

The Budding phenomenon. Cell membrane not affected

[43, 51, 52, 67, 74, 75, 77].

**4.4 Inflammation induced by radiation**

Action, Radiomimetic Properties and Possible Countermeasures for Radiation Injury 233

**Apoptosis Necrosis**

Caspase activation always present. Caspase depended initiation of developing

Genetic control initiate apoptosis No genetic control, environmentally-

Table 7. Apoptosis and Necrosis: Comparison of Morphological and Biochemical features. However, apoptosis is the principal mechanisms by which cells are physiologically eliminated. During apoptotic death, cellular dissociation by caspases of cell compartments occur and cell compartments and molecules are packaged into apoptotic bodies to avoid immune activation. [24,34,78]. Necrosis was considered in past times as passive, uncontrolled and unorganized mechanism of cell death. At the present time necrosis is considered as an alternative form of programmed cell death whose activation induces important biological consequences including immune reactions and induced inflammation.

Radiation induces an important and aggressive inflammatory response in the irradiated tissues. [19, 20, 28, 30, 64, 65]. Acute Radiation Disease (ARD) or Acute Radiation Syndromes (ARS) were defined as a toxic, poisonous exposure with development of the acute pathological processes in irradiated animals: systemic inflammatory response syndrome (SIRS), toxic multiple organ injury (TMOI), toxic multiple organ dysfunction syndromes (TMOD), toxic multiple organ failure (TMOF) [2, 5, 18, 19, 21, 89, 90, 91]. Radiation-induced lung injury, radiation pneumonitis (RP), is a potentially fatal side-effect of thoracic radiation therapy [3, 4, 11, 15]. Radiation-induced thyroiditis: can occur after the use of radiation treatment and can result in lifelong hypothyroidism [18]. Irradiation of the liver with doses of 2,450-2,920 rads caused liver injury in 14 patients of 65 patients treated with radiation therapy [3, 28]. The clinical picture of radiation-induced hepatitis includes hepatomegaly, ascites, pleural effusion and alteration in liver function. Radiation injury of the central nervous system (CNS) has devastating and fatal clinical consequences and

affected.

induced

Morphologically cells swelling, lysis. Mitochondria's structure and functions are

Karyolysis, Pyknosis, Karyorhesis. Nuclear swelling, Chromatin granular, Chromatin flocculation, Types of radiation induced damage of DNA: Breaks of the strand, alteration to bases, destruction of sugar, cross-links and formation of dimmers

of necrosis possible. Caspase-8 initiation possible with apoptosis and necrosis.

The Blebbing phenomenon, increased membrane's permeability. Cell membrane affected by external and internal factors.

Inflammation never present. Inflammation always present. Toxic substances never present. Toxic substances always present.

Table 6. Venom toxins and mechanisms of action – biochemical and clinical classification.

cysteine proteases, cleavage of PARP - poly (ADP-ribosyl) ation of nuclear proteins causes the inner mitochondrial membrane to become permeable for ions and other small molecules during both types of cell death- apoptosis and necrosis. Many mechanisms of cell necrosis and apoptosis are identical – for example, degradation of the Nuclear matrix is a Common Element during Radiation-Induced Apoptosis and Necrosis [1,3,11,16,24,34,38,43,53,59,78]. Data from different sources described that the differences between apoptosis and necrosis are much less numerous than previously studies suggested. [1,78]. Joseph Dynlacht and his research group of scientists studied the outcome of human promyelocytic leukemia (HL60) cells irradiated with 10 or 50 Gy of X-rays and determined the mode of leukemic cell death. Different doses of X-rays induced different modes of cell death – cells irradiated with 10 Gy died by necrosis; cells irradiated with 50Gy died predominantly by apoptosis. [16].

The Cell Death Nomenclature Committee recommends the use of the appropriate diagnosis: apoptotic necrosis. [34]. Comparison and description of a cell's Necrosis and Apoptosis were provided by a group of scientists from different institutions and research groups [16,34]. Apoptosis is described as a normally active, programmed process of cell death devoid of inflammation and toxins. [16,34]. Necrosis possesses the characteristics of accidental or externally-induced cell death from the influence of different environmental factors and triggering development of inflammation and elaborating of toxins, release of proinflammatory active and toxic cellular contents into the intracellular and extracellular fluids and into the lymphatic systems and blood circulation [39,55,56,62]. Caspases are important molecular mechanisms and central components of the apoptosis process [39, 50, 51, 52, 54, 58, 62, 74, 75, 77, 78].

myolytic, procoagulant, cardiotoxic, necrotic.

Table 6. Venom toxins and mechanisms of action – biochemical and clinical classification.

died by necrosis; cells irradiated with 50Gy died predominantly by apoptosis. [16].

The Cell Death Nomenclature Committee recommends the use of the appropriate diagnosis: apoptotic necrosis. [34]. Comparison and description of a cell's Necrosis and Apoptosis were provided by a group of scientists from different institutions and research groups [16,34]. Apoptosis is described as a normally active, programmed process of cell death devoid of inflammation and toxins. [16,34]. Necrosis possesses the characteristics of accidental or externally-induced cell death from the influence of different environmental factors and triggering development of inflammation and elaborating of toxins, release of proinflammatory active and toxic cellular contents into the intracellular and extracellular fluids and into the lymphatic systems and blood circulation [39,55,56,62]. Caspases are important molecular mechanisms and central components of the apoptosis process [39, 50, 51, 52, 54,

cysteine proteases, cleavage of PARP - poly (ADP-ribosyl) ation of nuclear proteins causes the inner mitochondrial membrane to become permeable for ions and other small molecules during both types of cell death- apoptosis and necrosis. Many mechanisms of cell necrosis and apoptosis are identical – for example, degradation of the Nuclear matrix is a Common Element during Radiation-Induced Apoptosis and Necrosis [1,3,11,16,24,34,38,43,53,59,78]. Data from different sources described that the differences between apoptosis and necrosis are much less numerous than previously studies suggested. [1,78]. Joseph Dynlacht and his research group of scientists studied the outcome of human promyelocytic leukemia (HL60) cells irradiated with 10 or 50 Gy of X-rays and determined the mode of leukemic cell death. Different doses of X-rays induced different modes of cell death – cells irradiated with 10 Gy

Induce two forms of

some neurons.

programmed cell death: cell apoptosis or cell necrosis. Presynaptic or postsynaptic actions. Induce excitation of

Neurotoxicity, Haemototoxity.

 Neurotoxicity. Haemototoxicity. Skin damage.

cardiovascular toxicity.

Hair follicule damage.

Anticoagulant proteins. Platelet aggregation inducers. Platelet aggregation inhibitors.

Induce cerebrovascular toxicity,

**Venom Toxins Mechanism of Action Symptoms**

Neurotoxic, Cardiotoxic, Myotoxic

Disorder of haemostasis, necrotic, haemolytic.

Cardiotoxic, Neurotoxic.

Polypeptide toxins. Autonomic,

Enzyme toxins: Transferases, Oxydoreductases, Hydrolases, Carbonnitrogen lyases, Metalloproteinases.

The Group of Toxins: Lectins, Nerve growth factors, Phospholipase inhibitors, Complement inhibitors, Proteinase

inhibitors.

58, 62, 74, 75, 77, 78].

Phospholipase toxins. Presynaptic neurotoxic,

Table 7. Apoptosis and Necrosis: Comparison of Morphological and Biochemical features.

However, apoptosis is the principal mechanisms by which cells are physiologically eliminated. During apoptotic death, cellular dissociation by caspases of cell compartments occur and cell compartments and molecules are packaged into apoptotic bodies to avoid immune activation. [24,34,78]. Necrosis was considered in past times as passive, uncontrolled and unorganized mechanism of cell death. At the present time necrosis is considered as an alternative form of programmed cell death whose activation induces important biological consequences including immune reactions and induced inflammation. [43, 51, 52, 67, 74, 75, 77].

## **4.4 Inflammation induced by radiation**

Radiation induces an important and aggressive inflammatory response in the irradiated tissues. [19, 20, 28, 30, 64, 65]. Acute Radiation Disease (ARD) or Acute Radiation Syndromes (ARS) were defined as a toxic, poisonous exposure with development of the acute pathological processes in irradiated animals: systemic inflammatory response syndrome (SIRS), toxic multiple organ injury (TMOI), toxic multiple organ dysfunction syndromes (TMOD), toxic multiple organ failure (TMOF) [2, 5, 18, 19, 21, 89, 90, 91]. Radiation-induced lung injury, radiation pneumonitis (RP), is a potentially fatal side-effect of thoracic radiation therapy [3, 4, 11, 15]. Radiation-induced thyroiditis: can occur after the use of radiation treatment and can result in lifelong hypothyroidism [18]. Irradiation of the liver with doses of 2,450-2,920 rads caused liver injury in 14 patients of 65 patients treated with radiation therapy [3, 28]. The clinical picture of radiation-induced hepatitis includes hepatomegaly, ascites, pleural effusion and alteration in liver function. Radiation injury of the central nervous system (CNS) has devastating and fatal clinical consequences and

Radiation Toxins – Effects of Radiation Toxicity, Molecular Mechanisms of

toxicity, as is attempted to be summarized in this chapter.

Carlos Montesinos, Kedar Prasad, Michael Epperly, Joel Greenberger.

**5. Acknowledgements**

the ARS is poorly understood.

Action, Radiomimetic Properties and Possible Countermeasures for Radiation Injury 235

organelles and molecules, which can result in cellular dysfunction, mutation of nucleic acids, or even apoptotic cell death [107]. Currently there are no proven countermeasures for these exposures, aside from a clinical agent, Amifostine, (Etyhol™), which is used to reduce

The cytotoxic effects of different types of radiation may be the single most important clinicopathologic process by which oxidative damage is induced from reactive oxygen species and radiation toxicity induced by radiation toxins. Radiation toxins (SRDs) with high enzymatic activity and their ability to degraded a wide variety of extracellular proteins, lipids, carbohydrates and DNA molecules, induce damage of important intracellular compartments such as mitochondria, ion channels, DNA, as well as activating degradation of peptide bonds in important polypeptides in tissues and vascular endothelium. Yet the exact mechanism by which radiation toxins stimulate development of

Countermeasures against radiation toxicity can be developed both in oral formulas [109] and parenteral agents, e.g. Manganese SuperOxide Dismutase (MnSOD)-liposomes [108, 109] which can reduce radiation exposure-induced biological effects. The oral formulas consist of a combination of antioxidants and chemopreventive, anti-progression agents, which can reduce the likelihood of mild toxicity in acute low dose radiation exposures and tumor induction in chronic low dose exposures. The combination of oral formulas with higher dose reactive oxygen species scavengers, like Mn-SOD can improve survival in animals exposed to acute high dose total body irradiation [109]. In addition to these measures, active immunization by low, non-toxic doses of radiation toxins, (SRDs) can also be employed to reduce radiation toxicity. [104, 110] To be effective, SRD immunization must occur no less than 30 days before irradiation, and can be effective up to three years or more. Active immunization by radiation toxins can significantly improve the survival rate (up to 60%) versus placebo-controlled irradiated animals. Our studies attempt to show the potential ability of specific antibodies to neutralize radiation toxins and thus substantially reduce the effects on radiation-induced neuro-, vascular, gastrointestinal, and hematopoietic toxicity. [104, 110] Antiradiation antibodies prevent the radiation-induced cytolysis of selected groups of cells that are sensitive to radiation. Anti-radiation antibodies derived from different phases of the Acute Radiation Syndrome can compete with and thus prevent cytolysis mediated by cytotoxic lymphocytes. The therapeutic benefit of neutralization of SRD radiation toxins could make hemopoetic stem cell transplantation more effective. Antiradiation vaccine and IgG antibodies have shown activity in animals against several different types of radiation include gamma, heavy ions, and neutron irradiation.[104, 110] Preliminary and pilot studies in vitro, in animal models and recently in humans, are showing some promise for both efficacy and safety/tolerability. [108-110] With current unpredictability in the operation of nuclear power facilities, potential radio-biologic terrorist weapons, and human operations in extreme environments, it is important to develop countermeasures to radiation toxicity in order to protect the potentially exposed. The first step in developing these countermeasures is to fully understanding the mechanism of

mucositis and other side effects from radiation therapy dose in cancer patients.

severely limits the dose that can be delivered in the radiotherapy of brain, head and neck, pulmonary and other tumors [19, 28, 30, 64, 65]. Hematopoetic system toxicity is a major limiting factor in the use of aggressive, combined modality therapy (chemotherapy + radiotherapy) in the treatment of malignant disease [14, 22, 40].

Permanent anemia and pancytopenia can be caused by the reduced capability for cellular proliferation due to the stem cells injury produced by radiation. [14, 69, 70, 72, 76]. Radiation induces an inflammatory response in the irradiated organs characterized by leukocyte infiltration and vascular permeability changes and vascular injury [90, 99]. Vascular injury is a key determinant of acute and chronic organ intoxication and dysfunction associated with irradiation of the different systems, including the gastrointestinal tract [9, 19, 20, 22,]. Radiotherapy is used in the treatment of pediatric brain tumors and is often associated with inflammation and behavior changing following irradiation-induced inflammation and injury in the developing brain [19, 20]. The consequences of irradiation microglial loss can be secondary to injury from pronounced inflammation [19, 20]. A prominent inflammatory response occurs following both ischemic and hemorrhagic forms of microcirculation disorders [19, 20].

#### **4.5 Roles of radiation toxins in developing of cellular structure radiation-induced injury**

Radiation Toxins contained a group of biologically-active compounds which induce complicated but specific mechanisms of action with tissues and cellular damage. Radiation Toxins include a group of glycoproteins with high enzymatic activity against proteins, lipids, carbohydrates. Radiation Toxins contain a group of peptides with selective affinity to number of receptors and specific interactions with receptors of mammalian organisms. Toxins isolated from the irradiated mammals inhibit or activate a group of an ion channels, acetylcholine receptors, nicotinic receptors, membranes, coagulant/anticoagulant pathways. Our experiments involved injections to radiation-naïve animals of toxic doses of Radiation Toxins (SDR- group) which damaged endothelial cells of blood and lymphatic vessels, and injury to blood microcirculation in the brain, lungs, gastrointestinal tract, cardiovascular system. Administration (IV or IM) of the Haemotopoietic Radiation Toxins to radiationnaïve animals induced development of anemia with reduction of hemoglobin concentration, reduction of red blood cell count and reduction of white cell blood count. Hemotopoietic Radiation Toxins induced the development of hemolytic anaemia with accelerated destruction of mature red cells outside the bone marrow and destruction of pluripotent stem cells inside of the bone marrow. Radiation-induced hemolytic anemia occurred with intravascular and extravascular hemolysis. As a result of stem cell injury erythropenia, leukopenia, thropmbocytopenia developed within days after irradiation. However, the acute hemolytic extravascular and intravascular anemia with lysis of erythrocytes, developed after high doses of radiation.

#### **4.6 Development of radiation countermeasures**

There are many sources of oxidative stress in the lives of workers, whether they work in nuclear power facilities, on the front lines of international conflicts, or in the reaches of outer space. The exposure dose can vary substantially, but at minimum will accelerate the aging of their organ systems, and at worse could result in acute exposure syndromes that may be fatal. A common thread of the oxidative stress exposures is ROS-binding to critical cellular

severely limits the dose that can be delivered in the radiotherapy of brain, head and neck, pulmonary and other tumors [19, 28, 30, 64, 65]. Hematopoetic system toxicity is a major limiting factor in the use of aggressive, combined modality therapy (chemotherapy +

Permanent anemia and pancytopenia can be caused by the reduced capability for cellular proliferation due to the stem cells injury produced by radiation. [14, 69, 70, 72, 76]. Radiation induces an inflammatory response in the irradiated organs characterized by leukocyte infiltration and vascular permeability changes and vascular injury [90, 99]. Vascular injury is a key determinant of acute and chronic organ intoxication and dysfunction associated with irradiation of the different systems, including the gastrointestinal tract [9, 19, 20, 22,]. Radiotherapy is used in the treatment of pediatric brain tumors and is often associated with inflammation and behavior changing following irradiation-induced inflammation and injury in the developing brain [19, 20]. The consequences of irradiation microglial loss can be secondary to injury from pronounced inflammation [19, 20]. A prominent inflammatory response occurs following both ischemic and hemorrhagic forms of microcirculation

**4.5 Roles of radiation toxins in developing of cellular structure radiation-induced**

Radiation Toxins contained a group of biologically-active compounds which induce complicated but specific mechanisms of action with tissues and cellular damage. Radiation Toxins include a group of glycoproteins with high enzymatic activity against proteins, lipids, carbohydrates. Radiation Toxins contain a group of peptides with selective affinity to number of receptors and specific interactions with receptors of mammalian organisms. Toxins isolated from the irradiated mammals inhibit or activate a group of an ion channels, acetylcholine receptors, nicotinic receptors, membranes, coagulant/anticoagulant pathways. Our experiments involved injections to radiation-naïve animals of toxic doses of Radiation Toxins (SDR- group) which damaged endothelial cells of blood and lymphatic vessels, and injury to blood microcirculation in the brain, lungs, gastrointestinal tract, cardiovascular system. Administration (IV or IM) of the Haemotopoietic Radiation Toxins to radiationnaïve animals induced development of anemia with reduction of hemoglobin concentration, reduction of red blood cell count and reduction of white cell blood count. Hemotopoietic Radiation Toxins induced the development of hemolytic anaemia with accelerated destruction of mature red cells outside the bone marrow and destruction of pluripotent stem cells inside of the bone marrow. Radiation-induced hemolytic anemia occurred with intravascular and extravascular hemolysis. As a result of stem cell injury erythropenia, leukopenia, thropmbocytopenia developed within days after irradiation. However, the acute hemolytic extravascular and intravascular anemia with lysis of erythrocytes,

There are many sources of oxidative stress in the lives of workers, whether they work in nuclear power facilities, on the front lines of international conflicts, or in the reaches of outer space. The exposure dose can vary substantially, but at minimum will accelerate the aging of their organ systems, and at worse could result in acute exposure syndromes that may be fatal. A common thread of the oxidative stress exposures is ROS-binding to critical cellular

radiotherapy) in the treatment of malignant disease [14, 22, 40].

disorders [19, 20].

developed after high doses of radiation.

**4.6 Development of radiation countermeasures**

**injury**

organelles and molecules, which can result in cellular dysfunction, mutation of nucleic acids, or even apoptotic cell death [107]. Currently there are no proven countermeasures for these exposures, aside from a clinical agent, Amifostine, (Etyhol™), which is used to reduce mucositis and other side effects from radiation therapy dose in cancer patients.

The cytotoxic effects of different types of radiation may be the single most important clinicopathologic process by which oxidative damage is induced from reactive oxygen species and radiation toxicity induced by radiation toxins. Radiation toxins (SRDs) with high enzymatic activity and their ability to degraded a wide variety of extracellular proteins, lipids, carbohydrates and DNA molecules, induce damage of important intracellular compartments such as mitochondria, ion channels, DNA, as well as activating degradation of peptide bonds in important polypeptides in tissues and vascular endothelium. Yet the exact mechanism by which radiation toxins stimulate development of the ARS is poorly understood.

Countermeasures against radiation toxicity can be developed both in oral formulas [109] and parenteral agents, e.g. Manganese SuperOxide Dismutase (MnSOD)-liposomes [108, 109] which can reduce radiation exposure-induced biological effects. The oral formulas consist of a combination of antioxidants and chemopreventive, anti-progression agents, which can reduce the likelihood of mild toxicity in acute low dose radiation exposures and tumor induction in chronic low dose exposures. The combination of oral formulas with higher dose reactive oxygen species scavengers, like Mn-SOD can improve survival in animals exposed to acute high dose total body irradiation [109]. In addition to these measures, active immunization by low, non-toxic doses of radiation toxins, (SRDs) can also be employed to reduce radiation toxicity. [104, 110] To be effective, SRD immunization must occur no less than 30 days before irradiation, and can be effective up to three years or more. Active immunization by radiation toxins can significantly improve the survival rate (up to 60%) versus placebo-controlled irradiated animals. Our studies attempt to show the potential ability of specific antibodies to neutralize radiation toxins and thus substantially reduce the effects on radiation-induced neuro-, vascular, gastrointestinal, and hematopoietic toxicity. [104, 110] Antiradiation antibodies prevent the radiation-induced cytolysis of selected groups of cells that are sensitive to radiation. Anti-radiation antibodies derived from different phases of the Acute Radiation Syndrome can compete with and thus prevent cytolysis mediated by cytotoxic lymphocytes. The therapeutic benefit of neutralization of SRD radiation toxins could make hemopoetic stem cell transplantation more effective. Antiradiation vaccine and IgG antibodies have shown activity in animals against several different types of radiation include gamma, heavy ions, and neutron irradiation.[104, 110]

Preliminary and pilot studies in vitro, in animal models and recently in humans, are showing some promise for both efficacy and safety/tolerability. [108-110] With current unpredictability in the operation of nuclear power facilities, potential radio-biologic terrorist weapons, and human operations in extreme environments, it is important to develop countermeasures to radiation toxicity in order to protect the potentially exposed. The first step in developing these countermeasures is to fully understanding the mechanism of toxicity, as is attempted to be summarized in this chapter.

## **5. Acknowledgements**

Carlos Montesinos, Kedar Prasad, Michael Epperly, Joel Greenberger.

Radiation Toxins – Effects of Radiation Toxicity, Molecular Mechanisms of

Radiology, 2005, p.1-6.

Sweden, 2009.

(2), pp.93 – 102.

Action, Radiomimetic Properties and Possible Countermeasures for Radiation Injury 237

transcription factors. Biochem Pharmacology Journal, 2002, 63, pp. 677-688. [18] Fliedner T. et al. Multi-organ involvement as a pathogenic principle of the radiation

[19] Kalm M. Inflammation and behavior following irradiation-induced injury in the

[20] Kleining T, Vink. R. Suppression of inflammation in ischemic and hemorrhagic stroke: therapeutic options. Current Opinion in neurology. 2009, No 22, pp. 294-301. [21] Konchalovsky M. et al. Multiple Organ Involvement and failure: selected Russian

[22] Swarts S. et al. Radiation-induced cyto-toxicity, DNA damage and DNA repair:

[23] Rakesh D. et al. Activation of a CrmA-insensitive, p35-sensitive pathway in Ionizing

[24] Fink S., Cookson B. Apoptosis, Pyroptosis and Necrosis: Mechanic Description of Dead and Dying Eukaryotic Cells. Infection and Immunity. Apr.2005, p.1907 – 1916. [25] Schultheiss, T., Kun, L., Ang, K. and Stephens, L. Radiation response of the central nervous system. *Int. J. Radiat. Oncol. Biol. Phys.* 31, 1093–1112 (1995). [26] Tofilon, P.J. and Fike, J.R. The radioresponse of the central nervous system: A dynamic

[27] Temple M., O'Leary D., Faden A. The role of glutamate receptors in the

[30] Van Der Kogel, A. Radiation-induced damage in the central nervous system: An interpretation of target cell responses. *Br. J. Cancer Suppl.* 7, 207–217 (1986). [31] Hungerford J. Committee on Natural Toxins and Food Allergens. General Referee

[32] Rudbach J., et al. Preparation and Properties of a National Reference Endotoxin. Journal

[33] Luvisetto S. et al. Toxicity of botulinum neurotoxins in central nervous system of mice.

[34] Levin S. Toxicological highlights: Apoptosis, Necrosis, or Oncosis: What is your

of Toxicologic Pathologists. Toxicological sciences. 1998, 41, 155 -156.

Diagnosis? A Report from the Cell Death Nomenclature Committee of the Society

Newcomb JK. John Wiley and Sons, Inc. New York. 2001. Pages 87-113. [28] Trot K. Radiation Risks from Imaging of Intestinal and Abdominal Inflammation. Scandinavian Journal of Gastroenterology. 1994, vol. 29, No 203, pp. 43-47. [29] Topchieva S., Mehrabova M., Influence of electromagnetic radiation on venom of

Vipera Lebetina Obtuse. Radioprotection, 2008, vol. 43, n. 5.

Reports: Journal of AOAC International. Vol., 89,No.1, 2006.

of Clinical Microbiology. Jan. 1976. P.21-25.

J.Toxicon, 2003, March, 41(4), pp.475 – 481.

pathophysiology of traumatic CNS injury. Chapter 4 in Head Trauma: Basic, Preclinical, and Clinical Directions. Miller LP and Hayes RL, editors. Co-edited by

Molecular Biology. 1997, v.272, No. 3, pp. 1965 – 1969.

process. *Radiation Research,* 153, 357–370 (2000).

(H(2)O(2)) caspase-3, and nuclear factor kappa-B (NF- kappa B), P53, c-Jun

syndromes: a study involving 110 cases histories documented in SEARCH and classified as the bases of hematopoietic indicators of effect. The British Institute of

developing brain. Center for Brain Repair and Rehabilitation, Institute of Neuroscience and Physiology, Sahlgrenska Academy, University of Gothenburg,

radiation accident cases re-visited. The British Institute of radiology. 2005,pp.26-29.

implications for cell survival theory. Radiation Environmental Biophysics. 1990, 29

Radiation-induced Apoptosis. The American Society for Biochemistry and

#### **6. References**

[1] Akadi Y, et al. Radiation-induced Apoptosis and Necrosis in Molt-4 Cells: A Study of

[2] Azizova T.V. et al. Multi-organ Involvement and failure in selected accident cases with

[3] Kumar A., Leeds N., Gregory N. et al. Malignant Gliomas: MR Imaging Spectrum of

[4] Ahmed, A. et al. Effect of Exposure to Electromagnetic Radiation from the Mobile Phone

[5] Berger M. and al. Medical management of radiation injuries: current approaches. Journal

[6] Tilmisanuy A., Osman A. Snake, Scorpion and Envenomation: A Review. JKAU: Med.

[7] Briggs S., Brinsfield K. Advanced Disaster Medical response Manual for Providers.

[8] Blakely W. Early biodosimetry response: recommendations for mass-casualty radiation

[9] Wong C.,Van der Kogel A. Mechanisms of Radiation injury to the Central Nervous

[10] Casarez E. Basic principles of toxicology. BIOC, 2001.p.597.Centers For Disease Control

[11] Chow, B., Li Y. and Wong, C. Radiation-induced apoptosis in the adult central nervous system is p53-dependent. Cell Death and Differentiation 7, 712–720 (2000). [12] Sears C, Kaper J. Enteric Bacterial Toxins: Mechanisms of Action and Linkage to

[13] Netto D., Chiacchio S. et al. Humoral response and neutralization capacity of sheep

[14] Daniak N, Ricks R. The evolving role of haemotopoetic cell transplantation in radiation

[15] Dewey W et al. Radiation induced apoptosis: relevance to radiotherapy. International Journal Radiation Oncology Biology Physics. 1995, No 33, pp. 781-796. [16] Dunlacht J., et al. Degradation of the Nuclear matrix Is a common Element during

[17] Rio D., Pardo V. Monoamine neurotoxins-induced apoptosis in lymphocytes by a

Intestinal Secretion. Microbiological Reviews, Mar. 1996, p.167-215.

accidents and terrorism. Armed Forces Radiobiology Research Institute, Uniformed

System: Indications for Neuroprotection. Molecular Interventions. 10. Volume 4,

and Prevention. Acute Radiation Syndrome: A fact Sheet for Physicians. June 29,

serum inoculated with natural and cobalt 60 – irradiated Crotalus durissus terrificus venom.LAURETTI, 1968. Journal of Venomous Animals and Toxins. Vol.

injury: potential and limitations. The British Institute of Radiology, 2005. Pp. 169-

Radiation-Induced Apoptosis and Necrosis. Radiation Research, 1999, v.152. No 6,

common oxidative stress mechanism: involvement of hydrogen peroxide

Adult Male Rats , Med. J. Cairo Univ*.*, 2006, 74(2),(Suppl.), 129–135.

Radiation Biology. 1993, V.64, No 1, pp. 47-56.

of Occupational Medicine. 2006. V.56. pp.162-172.

Harvard Medical International, Boston, Ma.2003,

Services University, Bethesda. pp. 1-26.

2005. www.bt.cdc.gov/radiation

Institute of Radiology.

Sci., Vol.3, pp. 11- 18.

Issue 5, 273-284, 2004.

8, No.2, 2002.

pp.590 – 603.

174.

Treatment.

Dose-effect Ralathionships and Their Modification. International Journal of

acute radiation syndrome observed at the Mayak Nuclear Facility. The British

Radiation Therapy- and Chemotherapy-induced Necrosis of the Brain after

on Acetylcholinestrase Activity in the Hippocampus and Striatum of Young and

**6. References**

(H(2)O(2)) caspase-3, and nuclear factor kappa-B (NF- kappa B), P53, c-Jun transcription factors. Biochem Pharmacology Journal, 2002, 63, pp. 677-688.

Radiation Toxins – Effects of Radiation Toxicity, Molecular Mechanisms of

Cancer Res Treat. 2007 March; 39(1): 16–21.

2009;8:155-181 ISSN 1611-2156

Cell Biology. 2004, v.5. p.897-907.

Pages 2753-2760.

104. Beijing,102205, China.

Harvard medical School.

Gastroenterology. 2007. Pp.3043-3046.

V.2010. article ID 234975. 12 pages.

Oncology Exchange. V.5. No 2. pp.29-31.

2003.

2010.

185–200. (Bios Scientific Publishers, Ltd. Oxford).

Action, Radiomimetic Properties and Possible Countermeasures for Radiation Injury 239

[53] Kim Y, M.D. Radiation-induced Necrosis Deteriorating Neurological Symptoms and

[54] Waterhouse, N.J. et al. (2003) Mitochondrial outer-membrane permeabilization in

[55] Verheij M. et al. Radiation-induced apoptosis. Cell Tissue Res. 2000, Jul; 301 (1): 133-42. [56] Rajesh P. et al. Apoptosis: molecular mechanisms and pathogenicity. EXCLI Journal

[57] Robertson J. and Orrenius S. Molecular Mechanisms of Apoptosis Induced by Cytotoxic

[58] Riedl S. et al.Molecular mechanisms of caspase regulations during apoptosis. Molecular

[59] Higuchi M. et al. Regulation of reactive oxygen species-induced apoptosis and necrosis

[60] Tan P, Khan A., Brusic V. Bioinformatics for Venom and Toxins Science. Henry Stewart

[61] Ji-Sheng C.. An Assessment of Toxins. Research Institute of Chemical Defense.P.O. Box

[62] Kathryn D. Held. Radiation Induced Apoptosis. Massachusetts General Hospital.

[63] Wirth B. and Olander D. Fundamentals of Radiation Damage and Defects Production.

[64] Molla M., Panes J. Radiation induced intestinal inflammation. World Journal of

[65] Molla M. et al. Influence of dose-rate on inflammatory damage and adhesion molecule

[66] Linard C., et al. Acute induction of inflammatory cytokine expression after gamma-

[67] Ramanan S. et al. Role of PPARs in Radiation-Induced Brain Injury. PPAR Research.

[68] Schiller T., Chen Y. Modeling radiation-induced lung injury risk with an ensemble of

[69] Baum S. et al. Radiation Induced Anemia in Rats Exposed repeatedly to mixed Gamma-

[70] Dritshilo A. Sherman D. Radiation and Chemical Injury in the Bone Marrow.

[71] Wiebe E., and Rodrigues G. Radiation Induced Lung Injury. Strategies for reducing

expression after abdominal radiation in the rat. International Journal of Radiation

irradiation in the rat: effect of an NF-kappa B inhibitor. International Journal of

support vector machines. Journal Neurocomputing , Volume 73. Issue 10-12, June,

damages while optimizing therapeutic doses. 2006. Parchurst. Publisher for

US Berkeley, Nuclear Engineering. Ref: Sect 17.2, D.O. text.

Oncology and Biological Physics. 1999.45. pp.1011-1018.

Radiation Oncology and Biological Physics. 2004.58. pp.427-434.

Neutron radiation. Radiation Research.1970. 41. pp.492-499.

Environment Health Prospectives. 1981, v.39, pp. 59-64.

by caspase 3-like proteases. Oncogene. 26 November 1998, Volume 17, Number 21,

Publications. 1467-5463. Briefings in Bioinformatics. Vol. 4. No 1. 53–62. March

Chemicals. Critical Reviews in Toxicology, 30(5):609–627 (2000).

Mimicking Progression of Brain Metastasis after Stereotactic-guided Radiotherapy.

apoptosis.In: Cell Proliferation and Apoptosis. Hughes, D. and Mehmet, H., eds.

[35] MacCann D. Radiation poisoning: Current concepts in the Acute Radiation Syndrome.

[36] Ramirez-Dominguez M, Olamendi T. et al. Cn11, the first example of a scorpion toxin

[37] Molla M, Panes J. Radiation-induced intestinal inflammation. World Journal of

[38] Guido M. and Isabelle J. Apoptosis, Oncosis, and Necrosis . An Overview of Cell Death.

[39] Ozsahin M. et al., CD4 and CD8 T-lymphocyte Apoptosis Can Predict Radiation-

[40] Xue M. at al. Intracortical Hemorrhage Injury in Rats : Relationship Between Blood

[41] Nieder C. et al. Experimental concepts for toxicity prevention and tissue restoration after central nervous system irradiation. Radiation Oncology. 2007, No 2, p.23. [42] Sagara-Ishijima N., Furuhama K. Toxic characteristics of the Synthetic Lipid A

[43] Nieder C., Andratschke N., Price R., Rivera, B., and Ang, K. Innovative prevention

[44] Gilles N., Blanchet C. et al. A Scorpion α-Like Toxin That Is Active on Insects and

[46] Wasalenko J, McVittie T et al. Medical Management of the Acute Radiation Syndrome:

[47] Bonner W. Low-dose radiation: Thresholds, bystander effects, and adaptive responses.

[48] UenoY . Toxicology of microbial toxins. Pure & Appl. Chem., Vol.58. No.2, pp.339-350.

[49] Yu-Cheng X, Song-Ping L. Inhibition of Sodium Channels In Rat Dorsal Root Ganglion

[50] Watters D. Molecular mechanisms of ionizing radiation-induced apoptosis.

[51] Haimovitz-Friedman A. Radiation-induced signal transduction and stress

[52] Szumiel I. Monitoring and signaling of radiation-induced damage in mammalian

that a true blocker of Na+ currents in crayfish neurons. The Journal of

Induced Late Toxicity: A Prospective Study in 399 Patients. Clinical Cancer

Fractions and Brain Cell Death. Journal of American Heart Association, 2000,

Derivative DT-5461 in Rats and Monkeys. Toxicological Sciences, 1999 ,49, 324-331.

strategies for radiation necrosis of the central nervous system. Anticancer Research*,*

Mammals Reveals an Unexpected Specificity and Distribution of Sodium Channel Subtypes in Rat Brain Neurons. The Journal of Neuroscience, October 15, 1999. [45] Vladimiriov A. Biological membranes and non-programmed cell death. Soros

Recommendations of the Strategic National Stockpile Radiation Working Group.

Proceeding of the national Academy of Sciences of United States of America. April

Neurons by Hainantoxin- IV, a novel Spider Toxin. Acta Biochimica et Biophysica

American Journal of Clinical Meidcine. V.3, No3. 2006, pp. 13-21.

Experimental Biology, 205, 869-876(2002).

Research, 2005, 11(20),October 15, 2005.

educational Journal. V.6, No 9, 2000, p.2-9.

2004. Ann Int Med. No. 140, pp.1037-1051.

Immunology and Cell Biology (1999) 77, 263–271.

response. Radiat. Res*.* 1998; 150: S102–8.

cells. Radiat. Res*.* 1998; 150: S92–101.

29, 2003, vol.10, no. 9, pp. 4973-4975.

No.31, pp.1721-1727.

22,1017–1023 .

1986.

Sinica, 2003, 35(1).

Gastroenterology, 2007, No 13, pp. 3043-3046.

American Journal of Pathology. 1995, v.146, No 1.

Radiation Toxins – Effects of Radiation Toxicity, Molecular Mechanisms of

15 July 2010, in Bremen, Germany, p.3.

Germany, p.3.

Cell Biol. 2004: 5, 752-762.

I/II, 2004. pp.85 -104.

P. 34.

Biosciences.

Action, Radiomimetic Properties and Possible Countermeasures for Radiation Injury 241

[89] Popov D., Maliev S. Antiradiation Vaccine: Immunological neutralization of Radiation

[90] Popov D., Maliev S. Radiation Toxins: Molecular mechanisms and radiomimetic

[91] Popov D., Maliev S. Countermeasure development : Specific Immunoprophylaxis and

[92] Jones, J. A., Riggs, P. K., Yang, T., Pedemonte, C. H., Clarke, M, Feedback, D., and Au,

injury and carcinogenesis ASEM 78(4) Section II suppl., pp. A67-78, 2007. [93] Koniarek J, Thomas J., Vazquez V. Detection of microlesions induced by heavy ions

[94] Koniarek J., Worgul B., Do heavy ions cause microlesions in cell membranes? Advances

[95] Zhivotovsky B., Samali, A., Gahm, A., and Orrenius, S Caspases: their intracellular

[96] Zhivotovsky, B., and Kroemer, G. Apoptosis and genomic instability. Nature Rev. Mol.

[97] Sorokina N.I. et al. Molecular mechanisms of radiation death of lymphoid cells. Protein poly-ADP-ribosylation in thymocyte fractions. Radiobiologiia. ;23 (6):739-42. [98] Filippovich I., Sorokina N. , Soldatenkov V., Alfyerova T., Trebenok Z. Effect of the

[99] Bebeshko V et al. Biological markers of ionizing radiation. Atomic Medicine. N1 , (39) –

[100] Popov D., Kirchin V., Maliev V., Zlobin V. All Union conference of agriculture

[101] Popov D., Kirchin V., Saphonova V, Usupova S. Republican Scientific Conference

[102] Gel Filtration, Principles and Methods Handbook, USA. 181-0221-8, Amersham,

[103] Maliev V., Kirchin V., Popov D. Antiradiation vaccine. Herald of VSC of Russian

[104] Maliev V., Popov D., Casey R, Jones J. Mechanisms of Action for an Anti-Radiation

culture model of motoneuron disease". BMC Neuroscience 10: 64.

Assembly. Held 18-15 July 2010, in Bremen, Germany, p.3.

in Space Research, Volume 12, Issues 2-3, 1992, Pages 417-420.

and Radiation Measurements in Low Earth Orbit.

Relat Stud Phys Chem Med. 1988 Apr. 53 (4):617-28.

radiobiology (USSR). Obninsk, 1990. P. 136–137.

Academy of Science. 2003. № 3. P. 12–28.

Toxins at Acute Radiation Syndromes. 38th COSPAR Scientific Assembly. Held 18-

properties. 38th COSPAR Scientific Assembly. Held 18-15 July 2010, in Bremen,

Immunotherapy of Combined Acute Radiation Syndromes. 38th COSPAR Scientific

W. Ionizing radiation-induced bioeffects in space and strategies to reduce cellular

using liposomes filled with fluorescent dye. Advances in Space Research , Volume 34, Issue 6, 2004, Pages 1373-1377. Space Life Sciences: Radiation Risk Assessment

localization and translocation during apoptosis. Cell Death Differ., 1999: 6, 644-651.

inducers of cellular differentiation and of ionizing radiation on thymus lymphocytes: chromatin degradation and programmed cell death. Int J Radiat Biol

"Theoretical and practical questions of veterinary medicine" (USSR). Kazan, 1990.

Vaccine in reducing the biological impact of high dose and dose-rate, low-linear energy transfer Radiation Exposure. J. Radiation Biology, v.47.N3. pp. 286-291. [105] Jaiswal M, Zech W, Goos M., Leutbecher C, Ferri A, Zippelius A, Carrì M, Nau R,

Keller BU (2009). "Impairment of mitochondrial calcium handling in a mtSOD1 cell

[72] Wei-Chung L. et al. Protection against Radiation-Induced Bone Marrow and Intestinal

[73] Wharton T. et al. Radiation Hepatitis induced by abdominal irradiation with the cobalt

[74] Elinger A., Thomson C. death by Design: Death by Design : Apoptosis, Necrosis and

[75] Dispersyn G. et al. Dissociation of cardiomyocite apoptosis and differentiation in infarct

[76] Ohgushi M. et al. Molecular Pathway and Cell State Responsible for Dissociation-

[77] Claudio A. et al. Caspase-dependent initiation of apoptosis and necrosis by the Fas

[78] Rastogi P. et al. Apoptosis: Molecular Mechanisms and Pathogenicity. EXCLI Journal

[79] Barnett N, editor. Biosecurity Reference: CFR-Listed Agent and Toxin Summaries. pp.1-

[80] Simon J. Nuclear, biological, and chemical terrorism: Understanding the threat and

[81] Franz D, DVM, PhD, Colonel, Veterinary Corps. United States Army. Defense against Toxin Weapons. U.S. Army Medical Research and Materiel Command, pp. 1-36. [82] Shivaji P. Snake venom neurotoxins: Pharmacological classification. Journal of

[83] Clemetson K, Morita T., Kini M. Scientific and standardization committee

[84] Viljorn C., Botes B. Snake Venom Toxins. The Journal of Biological Chemistry. Vol. 249,

[85] Oren B., Ph.D. Venom Toxins as Ion Channel Research Tools. Modulator No.21 Fall

[86] Alouf, J. et al. 1991. The family of mitogenic, shock-inducing and superantigenic toxins

[88] Gariepy, J., et al. 1987. Importance of disulfide bridges in the structure and activity of *Escherichia* coli enterotoxin ST1b. Proc. Natl. Acad. Sci. USA 84:8907–8911.

Sourcebook of bacterial protein toxins. Academic Press Ltd., London. [87] Fiorentini, C.,et al. 1995. Escherichia coli cytotoxic necrotizing factor 1: evidence for

communications: classification and nomenclature of snake venom C-type lectins

from staphylococci and streptococci, p. 367–414. *In* J. E. Alouf and J. H. Freer (ed.),

induction of actin assembly by constitutive activation of the p21 Rho GTPase.

designing responses. Int J Emergency Mental health 1999; 1: 81-89.

toxicology. Toxin reviews. 2004, vol. 23, no1, pp. 37-96 [60 page(s).

and related proteins. Posted on ISTH website,November 19, 2008.

No. 2, Issue of January 25, pp. 366-372, 1974.

2006. 28-31. www.alomone.com

Infect. Immun. 63:3936–3944.

Autophagy. Current Opinion in Cell Biology. 2004, pp. 653-669.

border zones. Europenean Heart Journal. 2002. 23. pp.849-857.

166, 900–907.

Ltd.

American Radium Society. 1973. pp.53-80.

August 6, 2010 Elsevier Inc.

published: August 03, 2009.

and Livermore, California 94550.

Injuries by *Cordyceps sinensis*, a Chinese Herbal Medicine. Radiation Research, 2006.

60 moving strip technique. Presented as the Fifty-fourth Annual Meeting of the

Induced Apoptosis in Human Pluripotent Stem Cells. Cell Stem Cell 7, 225–239,

receptor in lymphoid cells: onset of necrosis is associated with delayed ceramide increase. Journal of Cell Science 115, 4671-4683 © 2002 The Company of Biologists

2009;8:155-181 ISSN 1611-2156. Received: June 29, 2009, accepted: July 27, 2009,

150. Prepared by Sandia National Laboratory, Albuquerque, New Mexico 87185

**13**

*Mexico*

**Dynamical Aspects of Apoptosis**

*Autonomous University of Morelos, Cuernavaca, Morelos,*

*Theoretical and Computational Biology Group, Faculty of Science,*

Cells are dynamical systems characterized by a high degree of complexity due to an intricate network of subcellular processes that sustain their correct performance. This network is formed by a series of subnetworks with modular functioning, and among them is the subnetwork of biochemical processes that produces cell death or apoptosis. In this chapter we deal with the dynamical behavior of the apoptotic pathway activated by ionizing radiation (IR). In the special case of the apoptotic process, mathematical modeling is required for understanding its nonlinear complex behavior. Although previous attempts to model apoptosis were often limited to small systems, or based on qualitative data only, recent models are based on experimental data that give support to the theoretical results. In this form, mathematical models have shown that the interactions between the nuclear molecular components of the intrinsic apoptotic pathway drive the system into a persistent limit cycle in the respective phase space after the application of high doses of ionizing radiation. Experimental data confirm the existence of this limit cycle, showing that this dynamical feature of the system relies on the existence of a p53-Mdm2 negative feedback

At the cytoplasm, the molecular dynamics that leads to the activation of the caspases sets up a saddle point bifurcation in the *caspase 3-Bax* degradation rate diagram. In this region of bistability, the saddle point bifurcation can drive the system either to a stable point with a high concentration of caspase 3 (cell death) or to a stable point with a low concentration of caspase 3 (cell survival). Out of this region, the system dynamics is settled on by the existence of a single stable fix point, which determines cell surviving independently of the concentration of caspase 3, i.e., there exits a threshold mechanism that regulates apoptosis

Thus, the main goal of this chapter is to review and analyze the recent knowledge about these nonlinear dynamical aspects of IR-induced apoptosis, in order to understand how this complex subnetwork of molecular interactions integrates the information coded in the signals that control the cell survival, and executes the corresponding action to determine the correct death-life decision. The modeling of such decisions has a significant impact on the field of radiation biology and in the therapeutic aspects of inducing apoptosis in cancer cells by IR.

In recent decades cancer research has gained great importance worldwide, not only for being one of the most important chronic diseases of mankind, but also by the great

loop, which is induced by a switch-like activation of the ATM kinase.

and it is based on bistability, ultrasensitivity and irreversibility.

**2. Cell cycle and apoptosis**

**1. Introduction**

Antonio Bensussen and José Díaz

## **Dynamical Aspects of Apoptosis**

## Antonio Bensussen and José Díaz

*Theoretical and Computational Biology Group, Faculty of Science, Autonomous University of Morelos, Cuernavaca, Morelos, Mexico*

#### **1. Introduction**

242 Current Topics in Ionizing Radiation Research

[106] Falkow S. Molecular Koch's postulates applied to bacterial pathogenicity--a personal

[107] Jones J, Karouia F, Casey R. "Ionizing Radiation as a Carcinogen". In: McQueen C,

[108] Epperly, et al Modulation of Total Body Irradiation Induced Life Shortening by

[109] Epperly, M.W., Wang, H., Jones, J., Dixon, T., Montesinos, C.A., and Greenberger, J.S.

[110] Maliev, V, Popov,D, Jones, JA, Casey, RC. Mechanism of action for anti-radiation

[111] Baum S. Erythrocyte stem cell kinetics in postradiation rat. Armed Forces

[112] Mihai R., Doina G. Membrane structural and functional modifications due to oxidative

[113] Benderitter M. et al. The cell membrane as a biosensor of oxidative stress induced by

[114] Dixon D., Darveau R. Lipopolysaccharide Heterogenity: Innate Host Responses to Bacterial Modificationof Lipid A Structure. J. DenRes 84(7): 584-595, 2005. [115] Moroni M, Coolbaugh T., Lombardini E, Mitchell J, Moccia K, Shelton L, Nagy V,

[116] Kudriashov I., Goncharenko E, Deev L, Gorskaia T, Samoĭlikova T. Depression of

Atomic Physics. Report WP5 ID RAHAP 25-02/2002. pp. 1-8.

Res. 2011 Jul;176(1): 89-101. Epub 2011 Apr 26.

Research. 2009 January; 171(1): 41–52.

ed. Comprehensive Toxicology, Volume 14: Carcinogenesis. Oxford EN: Elsevier

Systemic Intravenous MnSOD-Plasmid Liposome Gene Therapy. Radiation

Antioxidant-Chemoprevention Diet Ameliorates Late Effects of Total Body Irradiation and Supplements Radioprotection by MnSOD-Plasmid Liposome

vaccine in reducing the biological impact of high-dose irradiation Advances in

Radiobiology Research Institute, Defense Atomic Support Agency, Bethesda,

stress induction by irradiation and chemical agents. European Commission Center of Excellence, InterDisciplinary research and Applications based on Nuclear and

radiation exposure: a multiparameter investigation.Radiation Research: vol.159, No

Whitnall M. Hematopoietic radiation syndrome in the Gottingen minipig. Radiat

endogenous radiation-sensitizing agents--lipid toxic substances--as one of the mechanisms of radiation protection. Dokl Akad Nauk SSSR. 1970; 195(1):206-8. [117] Smirnov D., Brady L., Halasa K, Morley M, Solomon S. and Cheun V.. Genetic variation in radiation induced cell death. Genome Research. August 15. 2011. [118] Sanchez M., Benitez A., Ortloff L., Green L. Alterations in Glutamate Uptake in NT2-

Derived Neurons and Astrocytes after Exposure to Gamma Radiation. Radiation

recollection 15 years later. Nat Rev Microbiol. 2004 Jan; 2(1):67-72.

Administration. Radiat. Res. 2011 June 175(6):759-765

2010

Research, 2008 170(4):437-444

Space Research 2007 40:586-590

Maryland. May. 1966.

4, pp.471-483.

Cells are dynamical systems characterized by a high degree of complexity due to an intricate network of subcellular processes that sustain their correct performance. This network is formed by a series of subnetworks with modular functioning, and among them is the subnetwork of biochemical processes that produces cell death or apoptosis. In this chapter we deal with the dynamical behavior of the apoptotic pathway activated by ionizing radiation (IR). In the special case of the apoptotic process, mathematical modeling is required for understanding its nonlinear complex behavior. Although previous attempts to model apoptosis were often limited to small systems, or based on qualitative data only, recent models are based on experimental data that give support to the theoretical results.

In this form, mathematical models have shown that the interactions between the nuclear molecular components of the intrinsic apoptotic pathway drive the system into a persistent limit cycle in the respective phase space after the application of high doses of ionizing radiation. Experimental data confirm the existence of this limit cycle, showing that this dynamical feature of the system relies on the existence of a p53-Mdm2 negative feedback loop, which is induced by a switch-like activation of the ATM kinase.

At the cytoplasm, the molecular dynamics that leads to the activation of the caspases sets up a saddle point bifurcation in the *caspase 3-Bax* degradation rate diagram. In this region of bistability, the saddle point bifurcation can drive the system either to a stable point with a high concentration of caspase 3 (cell death) or to a stable point with a low concentration of caspase 3 (cell survival). Out of this region, the system dynamics is settled on by the existence of a single stable fix point, which determines cell surviving independently of the concentration of caspase 3, i.e., there exits a threshold mechanism that regulates apoptosis and it is based on bistability, ultrasensitivity and irreversibility.

Thus, the main goal of this chapter is to review and analyze the recent knowledge about these nonlinear dynamical aspects of IR-induced apoptosis, in order to understand how this complex subnetwork of molecular interactions integrates the information coded in the signals that control the cell survival, and executes the corresponding action to determine the correct death-life decision. The modeling of such decisions has a significant impact on the field of radiation biology and in the therapeutic aspects of inducing apoptosis in cancer cells by IR.

#### **2. Cell cycle and apoptosis**

In recent decades cancer research has gained great importance worldwide, not only for being one of the most important chronic diseases of mankind, but also by the great

Dynamical Aspects of Apoptosis 245

it will interact with the ubiquitin ligase Mdm2 (Hdm2 in humans). Mdm2 binds to p53, forming a complex that eventually will be degraded in the proteosome (Jänicke et al., 2008). It is noteworthy that at the basal level the Mdm2 gene promoter is activated by transcription factors other than p53, which have not yet been identified (Phelps et al., 2003). It is known that the *Mdm2* gene has two promoters (P1 and P2) that can lead to at least two isoforms of Mdm2 (Candeias et al., 2008). The p90Mdm2 isoform is responsible of the inhibition of *p53*, and the p76Mdm2 isoform is a p53 activator (Perry, 2004) and promotes the translation of p53 mRNA (Candeias et al., 2008). Under basal conditions p76Mdm2 expression is greater than p90Mdm2 (Phelps et al., 2004), and this fact could explain the origin of the basal levels of p53 throughout the cell cycle (Figure 1a). However, under cellular stress conditions, p53 induces the expression of p90Mdm2 in detriment of the p76Mdm2 isoform (Perry, 2004).

Fig. 1. (a) **p53 network at basal conditions**. p53 and other transcription factors, some of them no yet identified and marked as X, activates *p53* gene. Part of the novel p53

repair signaling pathway, through activation of effector proteins like H2AX.

**2.3 Cellular response to genotoxic stress, and its different scenarios**

synthesized interacts with BCL-XL, forming heterodimers that will remain in the cytoplasm. The other part of p53 molecules is marked with an importation signal, enters the nucleus, and interacts with p90Mdm2 (represented by "Mdm2") forming a complex that degraded in the proteosome. "Y" represents a transcription factor set that activates p90Mdm2 in absence of genotoxic stress, and the empty set symbol represents protein degradation. (b) **ATM activation network in response to DNA damage.** DNA damage causes a nuclear topology change that induces the self-phosphorylation of ATM homodimers (represented by ATM/ATM), allowing its separation into active monomers (represented by ATM-on). The ATM activation is full after that TIP60 acetylates it. In the other side, if MRN complex (forming by Mre11, Rad50 and Nds1) detects broken DNA, it will be activated (represented by MRN-on) and then it will bind to broken DNA, in order to stop the damage propagation. After ATM activation, it will phosphorylate various substrates including Nbs1, and this permits ATM union to MRN-broken DNA complex, in order to start the DNA damage

The cell is able to respond to genotoxic stress produced by various endogenous and exogenous agents like ionizing radiation (IR), viral infection, heat, growth factor depravation and chemical DNA-damaging agents, among others (Plati et al., 2011). However, the mechanism and magnitude of the response will differ depending on both the

complexity of cellular process that it entails. A deeper insight into the causes of cancer progression, involves further research on the cell cycle because the malignant transformation starts as a deregulation of this cycle. In cancer, serious errors can be observed in DNA replication and repair simultaneously with an uncontrolled cell growth. Cancer cells lack the ability to respond to internal or external stimuli that induces death (Malumbres and Barbacid, 2009). It has seen that between 50 and 70% of all cancer types are related to the absence of the transcriptional factor p53 (Vousden, 2006), which serves to activate the cellular repairing response to DNA damage.

## **2.1 Regulation of the cell cycle progression**

The cell cycle has four phases. During the first phase (G1) the cell acquires the necessary nutrients for replication of the genetic material. If external signals are received, or the cell is ready to continue, then Cdc25 is activated (Budirahardja and Gönczy, 2009). Cdc25 activates Cdk4-cyclin-D and Cdk6-cyclin-D heterodimers (Malumbres, 2009). These heterodimers phosphorylate the Retinoblastoma protein (pRb) inducing the release of the E2F group of transcription factors (Abukhdeir and Park, 2009). These factors are necessary to transcript genes that commit the cell to pass from G1 to S phase (Abukhdeir and Park, 2009) and to activate the first checkpoint, where is evaluated the genetic material integrity (Abukhdeir and Park, 2009). This test is performed by some specialized proteins such as ATM kinase (Ataxia-Telangiectasia Mutated), as well as some DNA binding proteins like the MRN complex (Pommier and Weinstein, 2006). If these proteins detect DNA damage, they induce cell cycle arrest through effector molecules such as Chk2, and p21Waf1/cip1 (Abukhdeir and Park, 2009). If the damage is severe, apoptosis is activated through the p53 pathway. If not damage in the DNA is detected; the cell induces degradation of cyclin D, activating the Cdk2-cyclin-E heterodimer, which is necessary to start the S phase (Malumbres and Barbacid, 2009).

The checkpoint commanded by ATM remains active during the S phase, in order to ensure the integrity of cell genome (Budirahardja and Gönczy, 2009). At the end of the S phase, the Cdk2-cycline-A homodimer is activated to conclude the S phase, and the cell enters into G2 phase (Malumbres and Barbacid, 2009). After the cell has entered G2 phase, the activation of Cdk1-cyclin-B and Cdk1-cyclin-A heterodimers, in response to a mitogenic stimuli, starts M phase (Mitosis) (Pommier and Weinstein, 2006). It is noteworthy that the DNA damage sensors commanded by ATM are both active during transition from G2 to M and during the M phase (Calonge and O'Connell, 2008). After cytokinesis, cell can either enter to G1 phase or go out of cell cycle and remain in G0 state indefinitely (Calonge and O'Connell, 2008).

## **2.2 p53 network in stress-free conditions**

The p53 transcription factor essentially serves to regulate the expression of genes responsible for the cell cycle arrest, repair of DNA damage, senescence and apoptosis in the intrinsic pathway (Lacroix et al., 2006). It is known that expression of the gene encoding p53 is dependent of the p50 subunit of NF-kB transcription factor, and of other transcription factors like C/EPEβ-2, Ets-1, Pitx1, p73 and p53 itself (Baillat et al., 2006; Wang et al., 2006; Boggs and Reisman, 2007; Wu and Lozano, 1994). In stress-free human fibroblasts, p53 interacts with the anti apoptotic BCL-XL protein in the cytoplasm, and forms a heterodimer (Dogu & Díaz, 2009), and the remaining fraction of p53 is marked in the carboxyl-terminus with a nuclear import signal (NLS1), and is subsequently transported to the nucleus, where

complexity of cellular process that it entails. A deeper insight into the causes of cancer progression, involves further research on the cell cycle because the malignant transformation starts as a deregulation of this cycle. In cancer, serious errors can be observed in DNA replication and repair simultaneously with an uncontrolled cell growth. Cancer cells lack the ability to respond to internal or external stimuli that induces death (Malumbres and Barbacid, 2009). It has seen that between 50 and 70% of all cancer types are related to the absence of the transcriptional factor p53 (Vousden, 2006), which serves to

The cell cycle has four phases. During the first phase (G1) the cell acquires the necessary nutrients for replication of the genetic material. If external signals are received, or the cell is ready to continue, then Cdc25 is activated (Budirahardja and Gönczy, 2009). Cdc25 activates Cdk4-cyclin-D and Cdk6-cyclin-D heterodimers (Malumbres, 2009). These heterodimers phosphorylate the Retinoblastoma protein (pRb) inducing the release of the E2F group of transcription factors (Abukhdeir and Park, 2009). These factors are necessary to transcript genes that commit the cell to pass from G1 to S phase (Abukhdeir and Park, 2009) and to activate the first checkpoint, where is evaluated the genetic material integrity (Abukhdeir and Park, 2009). This test is performed by some specialized proteins such as ATM kinase (Ataxia-Telangiectasia Mutated), as well as some DNA binding proteins like the MRN complex (Pommier and Weinstein, 2006). If these proteins detect DNA damage, they induce cell cycle arrest through effector molecules such as Chk2, and p21Waf1/cip1 (Abukhdeir and Park, 2009). If the damage is severe, apoptosis is activated through the p53 pathway. If not damage in the DNA is detected; the cell induces degradation of cyclin D, activating the Cdk2-cyclin-E heterodimer, which is necessary to start the S phase (Malumbres and

The checkpoint commanded by ATM remains active during the S phase, in order to ensure the integrity of cell genome (Budirahardja and Gönczy, 2009). At the end of the S phase, the Cdk2-cycline-A homodimer is activated to conclude the S phase, and the cell enters into G2 phase (Malumbres and Barbacid, 2009). After the cell has entered G2 phase, the activation of Cdk1-cyclin-B and Cdk1-cyclin-A heterodimers, in response to a mitogenic stimuli, starts M phase (Mitosis) (Pommier and Weinstein, 2006). It is noteworthy that the DNA damage sensors commanded by ATM are both active during transition from G2 to M and during the M phase (Calonge and O'Connell, 2008). After cytokinesis, cell can either enter to G1 phase or go out of cell cycle and remain in G0 state indefinitely (Calonge and O'Connell, 2008).

The p53 transcription factor essentially serves to regulate the expression of genes responsible for the cell cycle arrest, repair of DNA damage, senescence and apoptosis in the intrinsic pathway (Lacroix et al., 2006). It is known that expression of the gene encoding p53 is dependent of the p50 subunit of NF-kB transcription factor, and of other transcription factors like C/EPEβ-2, Ets-1, Pitx1, p73 and p53 itself (Baillat et al., 2006; Wang et al., 2006; Boggs and Reisman, 2007; Wu and Lozano, 1994). In stress-free human fibroblasts, p53 interacts with the anti apoptotic BCL-XL protein in the cytoplasm, and forms a heterodimer (Dogu & Díaz, 2009), and the remaining fraction of p53 is marked in the carboxyl-terminus with a nuclear import signal (NLS1), and is subsequently transported to the nucleus, where

activate the cellular repairing response to DNA damage.

**2.1 Regulation of the cell cycle progression**

**2.2 p53 network in stress-free conditions**

Barbacid, 2009).

it will interact with the ubiquitin ligase Mdm2 (Hdm2 in humans). Mdm2 binds to p53, forming a complex that eventually will be degraded in the proteosome (Jänicke et al., 2008). It is noteworthy that at the basal level the Mdm2 gene promoter is activated by transcription factors other than p53, which have not yet been identified (Phelps et al., 2003). It is known that the *Mdm2* gene has two promoters (P1 and P2) that can lead to at least two isoforms of Mdm2 (Candeias et al., 2008). The p90Mdm2 isoform is responsible of the inhibition of *p53*, and the p76Mdm2 isoform is a p53 activator (Perry, 2004) and promotes the translation of p53 mRNA (Candeias et al., 2008). Under basal conditions p76Mdm2 expression is greater than p90Mdm2 (Phelps et al., 2004), and this fact could explain the origin of the basal levels of p53 throughout the cell cycle (Figure 1a). However, under cellular stress conditions, p53 induces the expression of p90Mdm2 in detriment of the p76Mdm2 isoform (Perry, 2004).

Fig. 1. (a) **p53 network at basal conditions**. p53 and other transcription factors, some of them no yet identified and marked as X, activates *p53* gene. Part of the novel p53 synthesized interacts with BCL-XL, forming heterodimers that will remain in the cytoplasm. The other part of p53 molecules is marked with an importation signal, enters the nucleus, and interacts with p90Mdm2 (represented by "Mdm2") forming a complex that degraded in the proteosome. "Y" represents a transcription factor set that activates p90Mdm2 in absence of genotoxic stress, and the empty set symbol represents protein degradation. (b) **ATM activation network in response to DNA damage.** DNA damage causes a nuclear topology change that induces the self-phosphorylation of ATM homodimers (represented by ATM/ATM), allowing its separation into active monomers (represented by ATM-on). The ATM activation is full after that TIP60 acetylates it. In the other side, if MRN complex (forming by Mre11, Rad50 and Nds1) detects broken DNA, it will be activated (represented by MRN-on) and then it will bind to broken DNA, in order to stop the damage propagation. After ATM activation, it will phosphorylate various substrates including Nbs1, and this permits ATM union to MRN-broken DNA complex, in order to start the DNA damage repair signaling pathway, through activation of effector proteins like H2AX.

#### **2.3 Cellular response to genotoxic stress, and its different scenarios**

The cell is able to respond to genotoxic stress produced by various endogenous and exogenous agents like ionizing radiation (IR), viral infection, heat, growth factor depravation and chemical DNA-damaging agents, among others (Plati et al., 2011). However, the mechanism and magnitude of the response will differ depending on both the

Dynamical Aspects of Apoptosis 247

In the nucleus, ATM interacts with NEMO forming a complex, which is ubiquitinated (Figure 2a), and transported to the cytoplasm where it induces the release of the transcription factor NF-kB (Schmid and Birbach, 2008). After that, NF-kB enters the nucleus and promotes the expression of various genes, including *BCL-XL* (Méndez et al., 2006; Chen et al., 2000). Simultaneously, ATM phosphorylates p53 nuclear stockpile, and, as a result, Mdm2 becomes less effective to recognize p53, delaying its degradation. Also, ATM phosphorylates Mdm2, inducing its autoubiquitination and degradation in the proteosome, and allowing an increase of p53 nuclear concentration over time (Dogu and Díaz, 2009). After ATM phosphorylation, Chk2 phosphorylates p53 in order to stabilize its structure (Pommier and Weinstein, 2006). At the end of this initial chain of phosphorylation, p53 can interact either with ASPP1 and ASPP2 proteins or with MUC1 (Pietsch et al., 2008). The MUC1 transmembrane glycoprotein is cleveaged in response to a genotoxic stimuli and the cytoplasmic segment is targeted to the nucleus, allowing its union with phosphorylated p53 and the induction of cell arrest (Pietsch et al., 2008). On the other hand, in response to severe DNA damage (Figure 2b), p53 will bind mainly with ASPP proteins. The concentration of the ASPP proteins increases gradually due to the activity of E2F-1 (Pietsch et al., 2008; Braithwaite, 2006), enhancing the pro-apoptotic function of p53. In this form, depending on the intensity of the stimulus, p53 can be oriented to transcribe pro-apoptotic genes or anti-

apoptotic genes, depending on the accessory proteins to which it is attached.

**2.4.2 Soft DNA damage, cell cycle arrest and successful repair of damage**

When the DNA damage signal is weak (Figure 3b), MUC1 nuclear concentration increases, and the union between p53 and nuclear MUC1 is enhaced. After that, the complex p53- MUC1 interacts with PCAF (p300/CBP associated factor) to form a new complex (Pietsch et al., 2008). PCAF acetylates p53 on lysine 320, allowing it to specifically recognize the promoters of genes related with the cell cycle arrest like p21WAF1/cip1, and DNA damage repair (Pietsch et al., 2008). Similarly, SIRT1 opposes PCAF-dependent phosphorylation of p53, blocking its transcriptional activity when the cell does not have energy to express proteins (Yamamori et al., 2009; Pediconi et al., 2009; Ikenoue et al., 2008). However, SIRT1 is not the only control point of p53. As discussed above, ATM, along with NEMO, activates NF-kB in its alternative pathway. This interaction is important, because NF-kB promotes the expression of anti-apoptotic genes like BCL-XL. NF-kB also competes with p53 for the cofactors: acetyltransferases p300, CBP and PCAF; which are required for the binding of p53

When DNA damage signal is either too strong (Figure 3a), or persistent, and if the cell has no energy to repair damage, p53 binds to ASPP proteins, and interacts with the histone acetyltransferase p300/CBP to form a complex (Patel et al., 2008). After that, p300/CBP acetylates p53 on lysine 373, allowing the selective recognization of the promoter sites of pro-apoptotic genes like *PUMA*, *BAX* and *BAK* by p53 (Pietsch et al., 2008). It is noteworthy that the formation of the p53-p300 complex is reversible, and p300 is separated from the complex by the removal of the acetyl group of p53 (Pediconi et al., 2009). The enzyme responsible for this step is the SIRT1 deacetylase (Tanno et al., 2006; Pediconi et al., 2009). SIRT1 is an enzyme that uses NAD+ as cofactor from cellular metabolism. However, in this scenario of severe DNA damage, the effect of SIRT1 is negligible, but will be important in

**2.4.1 Irreversible DNA damage, death by apoptosis**

the third scenario discussed below.

intensity and nature of the aggression. In particular, if the cell undergoes genotoxic stress there are three possible scenarios, and multiple combinations of them. The first scenario is when the cell suffers minor levels of DNA damage and can repair it successfully. The second scenario is when the cell suffers a medium damage in its DNA, and the cell has to "decide" between life and death. In this scenario it is possible that the cell die for not being able to fully repair the damage after a finite period of time, or live for being able to repair it. The third scenario is when the cell suffers severe damage, and initiates the apoptotic death. For a cell can be able of executing one of these three different responses, DNA damage intensity must be first detected, and then, the cell must "make the choice" of the appropriate scenario.

#### **2.4 Detection of DNA damage**

In genotoxic stress conditions, in response to DNA damage, the change in the nuclear topology, together with histone acetyl transferase Tip60, and the MRN complex (consisting of Mre11, Rad50, and Nbs1), induces the activation of ATM kinase (Sun et al., 2005; Vissers et al., 2008; Tanaka et al., 2007) (Figure 1b). After its activation, ATM phosphorylates histone H2AX and other proteins with a BRCT domain (such as Nbs1, 53BP1, and MDC1) for their activation (Vissers et al., 2008; Pommier and Weinstein, 2006). These proteins will bind to the broken DNA to stop the spread of damage (Vissers et al., 2008). Similarly, ATM activates Chk2 kinase, partially responsible for promoting cell cycle arrest, because it induces phosphorylation of Cdc25A and Cdc25C (Yoda et al., 2008; Pommier and Weinstein, 2006). Chk2 also promotes the activity of the transcription factor E2F-1 (Yoda et al., 2008), which, in turn, regulates Chk2 and ASPP protein expression (Bergamaschi et al., 2004).

Fig. 2. (a) **Module of ATM effector functions.** When ATM is anchored to MRN-broken DNA complex, ATM phosphorylates Chk2 inactive monomers, to promote its binding into active homodimers. After that, free active ATM with Chk2 active homodimers will phosphorylate Rb protein to release E2F1. This transcription factor controls the pro-apoptotic protein synthesis, such as ASPP proteins. Free active ATM can bind to NEMO to induce NF-kB release in the cytoplasm. It is important because NF-kB is responsible to regulate the expression of proteins required to cell survival, like BCL-XL. In figure the stepped line represents the induction of protein transcription. (b) **The p53 activation by ATM and Chk2.** After the activation of ATM and Chk2, ATM inhibits Mdm2. Together with Chk2, ATM phosphorylates p53 in order to stabilize it. Phosphorylated p53 can bind either to MUC1 to promote the transcription of genes related with the cell cycle arrest and DNA damage repair, or can bind to ASPP proteins to promote the transcription of genes related with apoptosis.

intensity and nature of the aggression. In particular, if the cell undergoes genotoxic stress there are three possible scenarios, and multiple combinations of them. The first scenario is when the cell suffers minor levels of DNA damage and can repair it successfully. The second scenario is when the cell suffers a medium damage in its DNA, and the cell has to "decide" between life and death. In this scenario it is possible that the cell die for not being able to fully repair the damage after a finite period of time, or live for being able to repair it. The third scenario is when the cell suffers severe damage, and initiates the apoptotic death. For a cell can be able of executing one of these three different responses, DNA damage intensity must be first detected, and then, the cell must "make the choice" of the appropriate scenario.

In genotoxic stress conditions, in response to DNA damage, the change in the nuclear topology, together with histone acetyl transferase Tip60, and the MRN complex (consisting of Mre11, Rad50, and Nbs1), induces the activation of ATM kinase (Sun et al., 2005; Vissers et al., 2008; Tanaka et al., 2007) (Figure 1b). After its activation, ATM phosphorylates histone H2AX and other proteins with a BRCT domain (such as Nbs1, 53BP1, and MDC1) for their activation (Vissers et al., 2008; Pommier and Weinstein, 2006). These proteins will bind to the broken DNA to stop the spread of damage (Vissers et al., 2008). Similarly, ATM activates Chk2 kinase, partially responsible for promoting cell cycle arrest, because it induces phosphorylation of Cdc25A and Cdc25C (Yoda et al., 2008; Pommier and Weinstein, 2006). Chk2 also promotes the activity of the transcription factor E2F-1 (Yoda et al., 2008), which, in

Fig. 2. (a) **Module of ATM effector functions.** When ATM is anchored to MRN-broken DNA complex, ATM phosphorylates Chk2 inactive monomers, to promote its binding into active homodimers. After that, free active ATM with Chk2 active homodimers will phosphorylate Rb protein to release E2F1. This transcription factor controls the pro-apoptotic protein synthesis, such as ASPP proteins. Free active ATM can bind to NEMO to induce NF-kB release in the cytoplasm. It is important because NF-kB is responsible to regulate the expression of proteins required to cell survival, like BCL-XL. In figure the stepped line represents the induction of protein transcription. (b) **The p53 activation by ATM and Chk2.** After the activation of ATM and Chk2, ATM inhibits Mdm2. Together with Chk2, ATM phosphorylates p53 in order to stabilize it. Phosphorylated p53 can bind either to MUC1 to promote the transcription of genes related with the cell cycle arrest and DNA damage repair, or can bind to ASPP proteins to

promote the transcription of genes related with apoptosis.

turn, regulates Chk2 and ASPP protein expression (Bergamaschi et al., 2004).

**2.4 Detection of DNA damage**

In the nucleus, ATM interacts with NEMO forming a complex, which is ubiquitinated (Figure 2a), and transported to the cytoplasm where it induces the release of the transcription factor NF-kB (Schmid and Birbach, 2008). After that, NF-kB enters the nucleus and promotes the expression of various genes, including *BCL-XL* (Méndez et al., 2006; Chen et al., 2000). Simultaneously, ATM phosphorylates p53 nuclear stockpile, and, as a result, Mdm2 becomes less effective to recognize p53, delaying its degradation. Also, ATM phosphorylates Mdm2, inducing its autoubiquitination and degradation in the proteosome, and allowing an increase of p53 nuclear concentration over time (Dogu and Díaz, 2009). After ATM phosphorylation, Chk2 phosphorylates p53 in order to stabilize its structure (Pommier and Weinstein, 2006). At the end of this initial chain of phosphorylation, p53 can interact either with ASPP1 and ASPP2 proteins or with MUC1 (Pietsch et al., 2008). The MUC1 transmembrane glycoprotein is cleveaged in response to a genotoxic stimuli and the cytoplasmic segment is targeted to the nucleus, allowing its union with phosphorylated p53 and the induction of cell arrest (Pietsch et al., 2008). On the other hand, in response to severe DNA damage (Figure 2b), p53 will bind mainly with ASPP proteins. The concentration of the ASPP proteins increases gradually due to the activity of E2F-1 (Pietsch et al., 2008; Braithwaite, 2006), enhancing the pro-apoptotic function of p53. In this form, depending on the intensity of the stimulus, p53 can be oriented to transcribe pro-apoptotic genes or antiapoptotic genes, depending on the accessory proteins to which it is attached.

#### **2.4.1 Irreversible DNA damage, death by apoptosis**

When DNA damage signal is either too strong (Figure 3a), or persistent, and if the cell has no energy to repair damage, p53 binds to ASPP proteins, and interacts with the histone acetyltransferase p300/CBP to form a complex (Patel et al., 2008). After that, p300/CBP acetylates p53 on lysine 373, allowing the selective recognization of the promoter sites of pro-apoptotic genes like *PUMA*, *BAX* and *BAK* by p53 (Pietsch et al., 2008). It is noteworthy that the formation of the p53-p300 complex is reversible, and p300 is separated from the complex by the removal of the acetyl group of p53 (Pediconi et al., 2009). The enzyme responsible for this step is the SIRT1 deacetylase (Tanno et al., 2006; Pediconi et al., 2009). SIRT1 is an enzyme that uses NAD+ as cofactor from cellular metabolism. However, in this scenario of severe DNA damage, the effect of SIRT1 is negligible, but will be important in the third scenario discussed below.

#### **2.4.2 Soft DNA damage, cell cycle arrest and successful repair of damage**

When the DNA damage signal is weak (Figure 3b), MUC1 nuclear concentration increases, and the union between p53 and nuclear MUC1 is enhaced. After that, the complex p53- MUC1 interacts with PCAF (p300/CBP associated factor) to form a new complex (Pietsch et al., 2008). PCAF acetylates p53 on lysine 320, allowing it to specifically recognize the promoters of genes related with the cell cycle arrest like p21WAF1/cip1, and DNA damage repair (Pietsch et al., 2008). Similarly, SIRT1 opposes PCAF-dependent phosphorylation of p53, blocking its transcriptional activity when the cell does not have energy to express proteins (Yamamori et al., 2009; Pediconi et al., 2009; Ikenoue et al., 2008). However, SIRT1 is not the only control point of p53. As discussed above, ATM, along with NEMO, activates NF-kB in its alternative pathway. This interaction is important, because NF-kB promotes the expression of anti-apoptotic genes like BCL-XL. NF-kB also competes with p53 for the cofactors: acetyltransferases p300, CBP and PCAF; which are required for the binding of p53

Dynamical Aspects of Apoptosis 249

Fig. 4. **Mdm2 and Wip1 down-regulation of p53 network**. The main targets of Wip1 are ATM, Chk2 and Mdm2 phosphorylated. This results in a silencing of the ATM signal, and the Mdm2 activation. Mdm2 inhibits p53, in order to return its levels to normality, and restored the network equilibrium. In figure red lines represent the interactions that blocks

The third stage is undoubtedly the most complicated of all, as it not only depends on the energy state of the cell, but also depends on the cell cycle phase in which the cell is. For example, if the cell is in S phase the cytoplasmic concentration of ASPP is higher than in other stages (Bermagaschi et al., 2004), and therefore this could lead more easily to apoptosis rather than cell cycle arrest. In this scenario, any type of delay suffered by the p53 dependent transcription process of the genes that control the DNA repair machinery and cell cycle arrest is of vital importance. These delays could be due to the effect of SIRT1 on p53, or to the competitive inhibition of p53 by other transcription factors, or to the action of the histone-acetyl transferases. If the transcription process of genes related to the DNA repair is slower than the normal, the cell will have fewer possibilities to survive. The intermittent signal of ATM produces a rise in the ASPP concentration, so it will be a question of time for the cell to enter in apoptosis. This could explain why after a long period of time if the cell is not repaired, it dies (Zhang et al., 2009; Yoda et al., 2001; Robinson et al., 2008; Shreeram et al., 2006; Wang et al., 2007; Bergamaschi et al., 2004; Braithwaite, 2006; Green and Kroemer, 2009; Toledo and Wahl, 2006; Liu et al., 2005; Haupt et al,, 2006; Strano

**2.4.3 Medium DNA damage, among life and death**

et al., 2006; Zhang et al., 2008).

Wip1.

to DNA (Jänicke et al., 2008). The effect of this interaction leads to the indirect repression of the transcriptional activity of p53, given the fact that the concentration of the cofactors remains constant (Schmid and Birbach, 2008; Youle and Strasser, 2008).

Fig. 3. (a) **The onset of apoptosis in its intrinsic pathway.** In response to severe DNA damage, p53 will bind to ASPP1 and ASPP2 (all of these are represented by ASPP). After that, p53-ASPP complex will interact with p300/CBP, and p300 acetylates p53-ASPP complex (represented by p53-ap). This allows p53-ap recognize the promoter side of apoptotic genes, like *PUMA* and *BAX*. The p53-ap transcriptional activity increases PUMA levels in the cytoplasm sufficiently to induce the release of p53 from p53/BCL-XL complexes. Free cytoplasmic p53 releases BAX in the cytoplasm starting C cytochromedependent apoptosis. (b) **Activation of cell cycle arrest and DNA damage repair pathways.**  In response to medium intensity DNA damage, p53 will bind to MUC1. The p53-MUC1 complex interacts with PCAF (represented by p53-cl) to allow the expression of genes related with cell cycle arrest like *p21*, as well as the expression of genes related with DNA repair (in the figure are represented by "Z"). Another remarkable function of p53-cl is inducing the Wip1 and Mdm2 expression. Wip1 is a phosphatase that down-regulates the p53 network.

Among the p53-target genes that are activated in this pathway are WIP1 phosphatase and Mdm2 in its p90 isoform. The Wip1 function is to dephosphorylate Chk2, ATM, activated p53 and Mdm2 (Figure 4). This leads to inactivation of all of these proteins except Mdm2, which is activated (Braithwaite, 2006; Candeias et al., 2008; Kobet et al., 2000; Lahav et al., 2004; Jänicke et al., 2008; Bernards, 2004). The dephosphorylated form of Mdm2, activated WIP1, and new synthesized Mdm2, inhibit p53 reducing its nuclear concentration (Hu et al., 2007). Because the WIP1 function, the signal generated by ATM and Chk2 will blink while DNA damage is not repaired, since ATM activator proteins remain linked to broken DNA (Jänicke et al., 2008; Lu et al., 2008). When DNA damage is repaired nuclear topology is restored, and ATM activating proteins trigger stop signal. In this case, Wip1 definitely inactivates ATM, and effector molecules like Chk2. This encourages Mdm2 (p90 isoform) to efficiently suppress p53 activity (Figure 4). These interactions allow the cell to return to its basal state, and the levels of p90, Mdm2 and Wip1 also return to their basal value (Bernards, 2004).

to DNA (Jänicke et al., 2008). The effect of this interaction leads to the indirect repression of the transcriptional activity of p53, given the fact that the concentration of the cofactors

Fig. 3. (a) **The onset of apoptosis in its intrinsic pathway.** In response to severe DNA damage, p53 will bind to ASPP1 and ASPP2 (all of these are represented by ASPP). After that, p53-ASPP complex will interact with p300/CBP, and p300 acetylates p53-ASPP complex (represented by p53-ap). This allows p53-ap recognize the promoter side of apoptotic genes, like *PUMA* and *BAX*. The p53-ap transcriptional activity increases PUMA

levels in the cytoplasm sufficiently to induce the release of p53 from p53/BCL-XL complexes. Free cytoplasmic p53 releases BAX in the cytoplasm starting C cytochromedependent apoptosis. (b) **Activation of cell cycle arrest and DNA damage repair pathways.**  In response to medium intensity DNA damage, p53 will bind to MUC1. The p53-MUC1 complex interacts with PCAF (represented by p53-cl) to allow the expression of genes related with cell cycle arrest like *p21*, as well as the expression of genes related with DNA repair (in the figure are represented by "Z"). Another remarkable function of p53-cl is inducing the Wip1 and Mdm2 expression. Wip1 is a phosphatase that down-regulates the

Among the p53-target genes that are activated in this pathway are WIP1 phosphatase and Mdm2 in its p90 isoform. The Wip1 function is to dephosphorylate Chk2, ATM, activated p53 and Mdm2 (Figure 4). This leads to inactivation of all of these proteins except Mdm2, which is activated (Braithwaite, 2006; Candeias et al., 2008; Kobet et al., 2000; Lahav et al., 2004; Jänicke et al., 2008; Bernards, 2004). The dephosphorylated form of Mdm2, activated WIP1, and new synthesized Mdm2, inhibit p53 reducing its nuclear concentration (Hu et al., 2007). Because the WIP1 function, the signal generated by ATM and Chk2 will blink while DNA damage is not repaired, since ATM activator proteins remain linked to broken DNA (Jänicke et al., 2008; Lu et al., 2008). When DNA damage is repaired nuclear topology is restored, and ATM activating proteins trigger stop signal. In this case, Wip1 definitely inactivates ATM, and effector molecules like Chk2. This encourages Mdm2 (p90 isoform) to efficiently suppress p53 activity (Figure 4). These interactions allow the cell to return to its basal state, and the levels of p90, Mdm2 and Wip1 also return to their basal value (Bernards,

p53 network.

2004).

remains constant (Schmid and Birbach, 2008; Youle and Strasser, 2008).

Fig. 4. **Mdm2 and Wip1 down-regulation of p53 network**. The main targets of Wip1 are ATM, Chk2 and Mdm2 phosphorylated. This results in a silencing of the ATM signal, and the Mdm2 activation. Mdm2 inhibits p53, in order to return its levels to normality, and restored the network equilibrium. In figure red lines represent the interactions that blocks Wip1.

## **2.4.3 Medium DNA damage, among life and death**

The third stage is undoubtedly the most complicated of all, as it not only depends on the energy state of the cell, but also depends on the cell cycle phase in which the cell is. For example, if the cell is in S phase the cytoplasmic concentration of ASPP is higher than in other stages (Bermagaschi et al., 2004), and therefore this could lead more easily to apoptosis rather than cell cycle arrest. In this scenario, any type of delay suffered by the p53 dependent transcription process of the genes that control the DNA repair machinery and cell cycle arrest is of vital importance. These delays could be due to the effect of SIRT1 on p53, or to the competitive inhibition of p53 by other transcription factors, or to the action of the histone-acetyl transferases. If the transcription process of genes related to the DNA repair is slower than the normal, the cell will have fewer possibilities to survive. The intermittent signal of ATM produces a rise in the ASPP concentration, so it will be a question of time for the cell to enter in apoptosis. This could explain why after a long period of time if the cell is not repaired, it dies (Zhang et al., 2009; Yoda et al., 2001; Robinson et al., 2008; Shreeram et al., 2006; Wang et al., 2007; Bergamaschi et al., 2004; Braithwaite, 2006; Green and Kroemer, 2009; Toledo and Wahl, 2006; Liu et al., 2005; Haupt et al,, 2006; Strano et al., 2006; Zhang et al., 2008).

Dynamical Aspects of Apoptosis 251

, , , , ,..., , ,... , ( , ) 1,2,..., *<sup>k</sup> k k s k k c t <sup>f</sup> c tc t c t c t D c t k s*

which is the well known form of the reaction-diffusion equation for the substance *k*. It indicates that the temporal variation of the concentration of *k* at point **x** at time *t* depends on the balance between the chemical processes in which this substance takes part, represented by the function *fk*, and its diffusion rate in the cellular medium. Function *fk* is generally a nonlinear function of the concentration of the reactive substances, and equation (3) usually has not an analytical solution. In a homogeneous medium the diffusive term in equation (3) is null, and *fk* completely defines the entire system dynamics in the *s*-dimensional phase space, which is defined by the set of concentration values of the *s* reactive substances. The systems dynamics is represented by a trajectory in this space, defined by the column vector **c**( )*t c tc t c t* 1 2 ... *<sup>s</sup>* . In nonlinear systems this trajectory can have peculiar characteristics like high sensitivity to initial conditions, bifurcations and complex loops that represent a great variety of dynamical behaviors observed in biological systems like limit cycles, hysteresis, bistability, ultrasensitivity, among others. In not homogeneous medium the diffusive term of equation (3) produces a more complex dynamical behavior of the system, giving rise to phenomena like traveling waves, spirals and spatially located bursting

In the subsequent sections it is discussed in depth the nonlinear dynamics of homogenous

The first problem concerning the dynamics of nonlinear systems is the determination the

becomes cero. Once the set of steady points of the system is settled on, is necessary to determine their stability. Equation (4) subject to the initial condition <sup>0</sup> **c c** 0 defines a

*o o* **c c** *t* **c c** , i.e., any trajectory that initiates at a distance

11 2

1 2

**c fc fc** (4)

The steady point *<sup>o</sup>* **c** of a dynamical system is **Liapunov stable** if for each

whenever 0

**c** remains at a distance

, where

*tt t*

*<sup>s</sup>* **c** *c tc t c t* for which equation:

*f ctct ct ct*

, ,... ,...

*k s*

> 0 exits a

> 0

*s k s*

of it for all positive time.

*f ctct ct ct*

, ,... ,...

*c tc t c t c t f c t c t c t c t*

**xx x x x x x x**

*rk r k s k k s*

1 2 1 2 , , ... , ... , , , , ,..., , ,... ,

**xx x x x** (3)

2

(2)

The reaction term of equation (1) can be rewritten as:

1 2

of second messengers and proteins, among others.

steady points of the system in its phase space.

A steady point is a column vector 1 2 ... *ooo o*

**3.1 Stability of nonlinear systems**

**nonlinear dynamical system**.

such that

of the steady point *<sup>o</sup>*

*r*

leading to:

1,2,...,

*t* **x**

*k s*

chemical systems.

#### **3. Nonlinear cell dynamics**

Cells are complex networks of physicochemical processes that support their highly organized structure and function. Thus, each cellular process sustained by a cell involves different levels of cellular organization. Each level of organization can be represented as a cell subsystem (subnetwork) that function in a modular mode **(**Thieffry and Romero, 1999; Del Vecchio, Ninfa and Sotang, 2008). In this form, cells can be modeled as formed by a set of subsystems like the regulatory genetic network, the network of synthesis and distribution of proteins, the network of signaling pathways, and the metabolic network, among others. The information flow through the set of cellular subsystems that controls the cell response to environmental signals occurs according to the canonical schema of Figure 5. Tumorinducing DNA or RNA viruses, and other agents like ionizing radiation (IR), can drastically modify this flow of information.

Fig. 5. **Flow of information between cell subsystems**. Changes in environmental conditions are sensed by cell signaling networks, which code and transmit this information to the nucleus. Coded information is decoded by the transcription machinery that, in response to this information, activates and inactivates a set of specific genes, giving raise to a specific distribution of effector proteins. These proteins are responsible for the cell specific response to the environmental conditions. Some of these proteins can be used to regulate gene expression, acting as specific transcription factors that conform complex regulatory loops inside the nucleus. Proteins can also be secreted to modify the cell environment. Tumorinducing viral RNA and DNA can modify this flow of information by acting directly on the cell genome, drastically modifying the population of effector proteins and their function. Ionizing radiation (IR) can drastically interrupt this flow of information by producing severe damage to the cell DNA, and inducing apoptosis.

An important remark from Figure 5 is that the flow of information through cellular subsystems is due to a continuous flow of matter and energy, according to the respective laws of conservation. Taking into account the flow of matter at each point of the cell, the respective mass balance equation is:

$$\frac{\partial c\_{k}(\mathbf{x},t)}{\partial t} = \sum\_{r} \nu\_{rk} \cdot a\_{r} \left( c\_{1}(\mathbf{x},t) c\_{2}(\mathbf{x},t) \dots c\_{k}(\mathbf{x},t) \dots c\_{s}(\mathbf{x},t) \right) + D\_{k} \nabla^{2} c\_{k}(\mathbf{x},t) \tag{1}$$

which means that the local rate of variation of the concentration of the substance *k* (denoted by *<sup>k</sup> c* ) at point **x** at time *t* is equal to the net rate of diffusion of *k* inside the cell volume *V* (denoted by <sup>2</sup> , *Dc t k k* **x** ) plus the rate of formation/degradation of *k* due to the local chemical reactions inside *V*. In equation (3.1), *<sup>r</sup>* represents the local rate of the chemical reaction *r*, which is a functional of the concentration of the reactive substances at point **x** at time *t*, and *rk* is the stoichiometric coefficient of *k* in reaction *r*.

The reaction term of equation (1) can be rewritten as:

$$\begin{aligned} \sum\_{\mathbf{x}} \nu\_{ik} \cdot a\_r \left( c\_1(\mathbf{x}, t) c\_2(\mathbf{x}, t) \dots c\_k(\mathbf{x}, t) \dots c\_s(\mathbf{x}, t) \right) &= f\_k \left( c\_1(\mathbf{x}, t), c\_2(\mathbf{x}, t), \dots, c\_k(\mathbf{x}, t), \dots c\_s(\mathbf{x}, t) \right) \\ \vdots &= 1, 2, \dots, s \end{aligned} \tag{2}$$

leading to:

250 Current Topics in Ionizing Radiation Research

Cells are complex networks of physicochemical processes that support their highly organized structure and function. Thus, each cellular process sustained by a cell involves different levels of cellular organization. Each level of organization can be represented as a cell subsystem (subnetwork) that function in a modular mode **(**Thieffry and Romero, 1999; Del Vecchio, Ninfa and Sotang, 2008). In this form, cells can be modeled as formed by a set of subsystems like the regulatory genetic network, the network of synthesis and distribution of proteins, the network of signaling pathways, and the metabolic network, among others. The information flow through the set of cellular subsystems that controls the cell response to environmental signals occurs according to the canonical schema of Figure 5. Tumorinducing DNA or RNA viruses, and other agents like ionizing radiation (IR), can drastically

Fig. 5. **Flow of information between cell subsystems**. Changes in environmental conditions are sensed by cell signaling networks, which code and transmit this information to the nucleus. Coded information is decoded by the transcription machinery that, in response to this information, activates and inactivates a set of specific genes, giving raise to a specific distribution of effector proteins. These proteins are responsible for the cell specific response to the environmental conditions. Some of these proteins can be used to regulate gene expression, acting as specific transcription factors that conform complex regulatory loops inside the nucleus. Proteins can also be secreted to modify the cell environment. Tumorinducing viral RNA and DNA can modify this flow of information by acting directly on the cell genome, drastically modifying the population of effector proteins and their function. Ionizing radiation (IR) can drastically interrupt this flow of information by producing severe

An important remark from Figure 5 is that the flow of information through cellular subsystems is due to a continuous flow of matter and energy, according to the respective laws of conservation. Taking into account the flow of matter at each point of the cell, the

> , , , ... , ... , ( ,) *<sup>k</sup> rk r k s k k*

which means that the local rate of variation of the concentration of the substance *k* (denoted by *<sup>k</sup> c* ) at point **x** at time *t* is equal to the net rate of diffusion of *k* inside the cell volume *V* (denoted by <sup>2</sup> , *Dc t k k* **x** ) plus the rate of formation/degradation of *k* due to the local

reaction *r*, which is a functional of the concentration of the reactive substances at point **x** at

*c tc t c t c t D c t*

**xx x x x** (1)

*<sup>r</sup>* represents the local rate of the chemical

 <sup>2</sup>

1 2

*rk* is the stoichiometric coefficient of *k* in reaction *r*.

**3. Nonlinear cell dynamics**

modify this flow of information.

damage to the cell DNA, and inducing apoptosis.

*r*

chemical reactions inside *V*. In equation (3.1),

respective mass balance equation is:

*c t*

*t* **x**

time *t*, and

$$\frac{\partial c\_k(\mathbf{x},t)}{\partial t} = f\_k\left(c\_1(\mathbf{x},t), c\_2(\mathbf{x},t), \dots, c\_k(\mathbf{x},t), \dots c\_s(\mathbf{x},t)\right) + D\_k \nabla^2 c\_k(\mathbf{x},t) \quad k = 1,2,\dots,s \tag{3}$$

which is the well known form of the reaction-diffusion equation for the substance *k*. It indicates that the temporal variation of the concentration of *k* at point **x** at time *t* depends on the balance between the chemical processes in which this substance takes part, represented by the function *fk*, and its diffusion rate in the cellular medium. Function *fk* is generally a nonlinear function of the concentration of the reactive substances, and equation (3) usually has not an analytical solution. In a homogeneous medium the diffusive term in equation (3) is null, and *fk* completely defines the entire system dynamics in the *s*-dimensional phase space, which is defined by the set of concentration values of the *s* reactive substances. The systems dynamics is represented by a trajectory in this space, defined by the column vector **c**( )*t c tc t c t* 1 2 ... *<sup>s</sup>* . In nonlinear systems this trajectory can have peculiar characteristics like high sensitivity to initial conditions, bifurcations and complex loops that represent a great variety of dynamical behaviors observed in biological systems like limit cycles, hysteresis, bistability, ultrasensitivity, among others. In not homogeneous medium the diffusive term of equation (3) produces a more complex dynamical behavior of the system, giving rise to phenomena like traveling waves, spirals and spatially located bursting of second messengers and proteins, among others.

In the subsequent sections it is discussed in depth the nonlinear dynamics of homogenous chemical systems.

#### **3.1 Stability of nonlinear systems**

The first problem concerning the dynamics of nonlinear systems is the determination the steady points of the system in its phase space.

A steady point is a column vector 1 2 ... *ooo o <sup>s</sup>* **c** *c tc t c t* for which equation:

$$\dot{\mathbf{c}}(t) = \mathbf{f}(\mathbf{c}(t))\_{\prime}, \text{ where } \mathbf{f}(\mathbf{c}(t)) = \begin{bmatrix} f\_1(c\_1(t), c\_2(t), \dots, c\_k(t), \dots, c\_s(t)) \\ \vdots \\ f\_s(c\_1(t), c\_2(t), \dots, c\_k(t), \dots, c\_s(t)) \end{bmatrix} \tag{4}$$

becomes cero. Once the set of steady points of the system is settled on, is necessary to determine their stability. Equation (4) subject to the initial condition <sup>0</sup> **c c** 0 defines a **nonlinear dynamical system**.

The steady point *<sup>o</sup>* **c** of a dynamical system is **Liapunov stable** if for each > 0 exits a > 0 such that whenever 0 *o o* **c c** *t* **c c** , i.e., any trajectory that initiates at a distance of the steady point *<sup>o</sup>* **c** remains at a distance of it for all positive time.

Dynamical Aspects of Apoptosis 253

2 2 2 2 2 2 1 2 1 2 12

*c t ct ct f f*

**J***c c* 1 2 , represents the Jacobian matrix of the linearized dynamical system in equation 6. The

arrangement of the trajectories near each steady point of the nonlinear system (Figure 6). This linearization process can be extended to perform the phase space analysis of higher

An important question that arises at this point is whether the behavior of the trajectories obtained from equation (7) accurately reflects the real behavior of the trajectories of the original nonlinear system. Andronov et al. (1973) show that this is the case. If the linearized system has a saddle, a node or a spiral at a given steady point, then the original nonlinear system really has also a saddle, a node or a spiral at that steady point. Furthermore, if a steady point is a stable saddle or node of the linearized dynamical system, then is also a stable saddle or node of the nonlinear system. In this case, the neglected nonlinear terms of equations (5) and (6) practically have not effect on the stability of these points when Re(

0 for both eigenvalues. This kind of steady points is known as **hyperbolic points**, and they are not affected by the small nonlinear perturbations. The topological implication of this fact is that the vector field corresponding to a saddle or a node is not altered by small nonlinear

a center. The trajectories around this point are closed orbits that are stable. However, the neglected nonlinear terms in equations (5) and (6) can produce an imperfect closure of the orbit, giving rise to a spiral. In this form, the vector field corresponding to a center is altered by small nonlinear perturbations that transform the center into a spiral and, as consequence,

According to their stability, steady points of 2-dimensional dynamical systems can be classified into a: 1) **Robust case**, which includes repellers or sources, for which both

**Marginal case**, which includes centers for which both eigenvalues are pure imaginary, and

However, the phase space of a nonlinear system can exhibit another kind of closed orbits called **limit cycles**, which cannot be observed in linear systems. A limit cycle is an isolated

) > 0; attractors or sinks, for which both eigenvalues have Re(

perturbations and, as consequence, has **structural stability** (Strogatz, 1994).

When the eigenvalues of the Jacobian matrix are pure imaginary

has not structural stability (Strogatz, 1994).

and saddles, for which one eigenvalues has Re(

non-isolated steady points for which one eigenvalue has Re(

eigenvalues have Re(

2 2 2 2 2

1 2 , , <sup>1</sup> 1 1

*c c c t ct ct*

*f f c c c c O c c cc*

of **J***c c* 1 2 , can be calculated from the characteristic equation:

. Depending on the nature of the eigenvalues, it is possible to know the

*c c*

,

1 2

) > 0 and the other one has Re() < 0; 2)

) = 0 (Strogatz, 1994).

= *i* , the steady point is

*c c* (6)

,,

**J** (7)

)

) < 0

*o o o o c c c c*

1 2 1 2

*cc cc*

1 2 1 2

*o o o o*

*cc cc*

1 2 , ,

1 2 1 2

*o o o o*

which leads to the linearized dynamical system:

1 2 ,

1 2 , ,

1 1

*f f*

*c c*

dimensional nonlinear dynamical systems (Edelstein-Kesher, 2005).

in a similar form:

eigenvalues

**J I** *c c* 1 2 , 0

The steady point *<sup>o</sup>* **c** of a dynamical system is attracting if exists a 0 such that lim *<sup>o</sup> <sup>t</sup>* **c c** *<sup>t</sup>* for any trajectory **c c** ( )*<sup>t</sup>* whenever 0 *<sup>o</sup>* **c c** , i.e., any trajectory that

initiates at a distance of the steady point *<sup>o</sup>* **c** will converge to it eventually. In this case, the point *<sup>o</sup>* **c** is an **attractor** of the dynamical system in the phase space. A steady point *<sup>o</sup>* **c** , which is Liapunov stable and attracting, is asymptotically stable.

A steady point *<sup>o</sup>* **c** , which is neither stable nor attracting, is unstable and is a **repulsor** in the phase space.

#### **3.2 Phase plane analysis**

In nonlinear systems the trajectories cannot be generally determined in an analytical form. However, it is possible to perform a qualitative analysis to find out the global behavior of the dynamical system in the corresponding phase space. As a vector can be assigned to each point of this space, according to equation (4), the vector field associated to the phase space can be drawn. By flowing this vector field, a phase point traces a solution **c**(*t*) of the dynamical system, corresponding to a trajectory winding trough the phase space.

It is of importance to point out the fact that if the function **f** of equation (4) is continuous and all its partial derivates , 1,2,..., *<sup>i</sup> j <sup>f</sup> ij s c* are also continuous in **c** for a given subset

 *<sup>n</sup> D* , then for every **c**<sup>0</sup> *D* the initial value problem of equation (4), has solution **c**(*t*) in some time interval (*-t,t*) around *t* = 0 and this solution is unique. A topological implication of this theorem is that two trajectories cannot intersect and, as consequence, chaos is ruled out of any 2-dimensional phase space but arises as a possible behavior of every sdimensional dynamical system with s > 2 (Strogatz, 1994).

The phase space analysis of the dynamics of a nonlinear system takes into account the following aspects: 1) the number, position and stability of the steady points; 2) the arrangement of the trajectories near the steady points; and 3) the existence and stability of closed orbits.

In section 3.1 was presented the form in which the steady points are determined and how they can be classified according to their stability. The arrangement of the trajectories around steady points is determined by linearization of the original nonlinear system, and analysis of the behavior of the eigenvalues of the Jacobian matrix of the linearized system around each steady point. For example, considering a 2-dimensional phase space and a steady point 1 1 , *o oo* **c** *c c* , a small perturbation from this steady state drives the nonlinear dynamical

system

$$
\begin{bmatrix}
\dot{c}\_1(t) \\
\dot{c}\_2(t)
\end{bmatrix} = \begin{vmatrix}
f\_1(c\_1(t), c\_2(t)) \\
f\_2(c\_1(t), c\_2(t))
\end{vmatrix} \text{into} \quad \text{a} \quad \text{new} \quad \text{trajector} \quad \delta \mathbf{c}(t) = \left\{\delta c\_1(t), \delta c\_2(t)\right\} \text{i}
$$

where 1 11 *<sup>o</sup> ct ct c* and <sup>222</sup> *<sup>o</sup> ct ct c* . In this form:

$$\begin{aligned} \delta \hat{c}\_1 &= \dot{c}\_1 = f\_1(c\_1^o, c\_2^o) + \frac{\partial f\_1}{\partial c\_1} \bigg|\_{\left(c\_1^{\prime}, c\_2^{\prime}\right)} \delta c\_1 + \frac{\partial f\_1}{\partial c\_2} \bigg|\_{\left(c\_1^{\prime}, c\_2^{\prime}\right)} \delta c\_2 + O\left(\delta^2 c\_1, \delta^2 c\_2, \delta \varepsilon\_1 \delta \varepsilon\_2\right) \\ &= \frac{\partial f\_1}{\partial c\_1} \bigg|\_{\left(c\_1^{\prime}, c\_2^{\prime}\right)} \delta c\_1 + \frac{\partial f\_1}{\partial c\_2} \bigg|\_{\left(c\_1^{\prime}, c\_2^{\prime}\right)} \delta c\_2 + O\left(\delta^2 c\_1, \delta^2 c\_2, \delta \varepsilon\_1 \delta \varepsilon\_2\right) \text{ because } f\_1(c\_1^o, c\_2^o) = 0 \end{aligned} \tag{5}$$

in a similar form:

252 Current Topics in Ionizing Radiation Research

point *<sup>o</sup>* **c** is an **attractor** of the dynamical system in the phase space. A steady point *<sup>o</sup>* **c** , which

A steady point *<sup>o</sup>* **c** , which is neither stable nor attracting, is unstable and is a **repulsor** in the

In nonlinear systems the trajectories cannot be generally determined in an analytical form. However, it is possible to perform a qualitative analysis to find out the global behavior of the dynamical system in the corresponding phase space. As a vector can be assigned to each point of this space, according to equation (4), the vector field associated to the phase space can be drawn. By flowing this vector field, a phase point traces a solution **c**(*t*) of the

It is of importance to point out the fact that if the function **f** of equation (4) is continuous and

 *<sup>n</sup> D* , then for every **c**<sup>0</sup> *D* the initial value problem of equation (4), has solution **c**(*t*) in some time interval (*-t,t*) around *t* = 0 and this solution is unique. A topological implication of this theorem is that two trajectories cannot intersect and, as consequence, chaos is ruled out of any 2-dimensional phase space but arises as a possible behavior of every s-

The phase space analysis of the dynamics of a nonlinear system takes into account the following aspects: 1) the number, position and stability of the steady points; 2) the arrangement of the trajectories near the steady points; and 3) the existence and stability of

In section 3.1 was presented the form in which the steady points are determined and how they can be classified according to their stability. The arrangement of the trajectories around steady points is determined by linearization of the original nonlinear system, and analysis of the behavior of the eigenvalues of the Jacobian matrix of the linearized system around each steady point. For example, considering a 2-dimensional phase space and a steady point 1 1 , *o oo* **c** *c c* , a small perturbation from this steady state drives the nonlinear dynamical

<sup>222</sup> *<sup>o</sup> ct ct c* . In this form:

, , because ( , ) 0

*f f <sup>c</sup> c O c c c c fcc*

*c c c c*

1 2 1 2

*o o o o*

*f f c c fcc <sup>c</sup> c O c c cc*

1 1 112 1 2 1 2 12 1 2 , ,

(,) , ,

*c c*

>

1 1 2 2

into a new trajectory

1 2 1 2 12 112

1 1 2 2

dynamical system, corresponding to a trajectory winding trough the phase space.

*<sup>f</sup> ij s*

*<sup>o</sup>* **c c** , i.e., any trajectory that

are also continuous in **c** for a given subset

 **c***t ct ct* 1 2 , ,

*o o*

(5)

of the steady point *<sup>o</sup>* **c** will converge to it eventually. In this case, the

0 such that

The steady point *<sup>o</sup>* **c** of a dynamical system is attracting if exists a

*<sup>t</sup>* **c c** *<sup>t</sup>* for any trajectory **c c** ( )*<sup>t</sup>* whenever 0

is Liapunov stable and attracting, is asymptotically stable.

all its partial derivates , 1,2,..., *<sup>i</sup>*

*j*

dimensional dynamical system with s > 2 (Strogatz, 1994).

*c c c c*

1 2 , ,

1 2 1 2

*o o o o*

*c c*

*o o*

, ,

1 11 2 2 21 2

*c t f ctct c t f ctct*

*c*

 lim *<sup>o</sup>*

phase space.

closed orbits.

system

where  1 11 *<sup>o</sup> ct ct c* and

initiates at a distance

**3.2 Phase plane analysis**

$$
\delta\dot{\mathcal{C}}\_2 = \dot{\mathcal{C}}\_2 = \frac{\partial f\_2}{\partial \mathbf{c}\_1}\bigg|\_{\left(\epsilon\_1^{\prime}, \epsilon\_2^{\prime}\right)} \delta \mathbf{c}\_1 + \frac{\partial f\_2}{\partial \mathbf{c}\_2}\bigg|\_{\left(\epsilon\_1^{\prime}, \epsilon\_2^{\prime}\right)} \delta \mathbf{c}\_2 + O\left(\delta^2 \mathbf{c}\_1, \delta^2 \mathbf{c}\_2, \delta \mathbf{c}\_1 \delta \mathbf{c}\_2\right) \tag{6}
$$

which leads to the linearized dynamical system:

$$
\begin{bmatrix}
\delta\dot{c}\_{1}(t) \\
\delta\dot{c}\_{2}(t)
\end{bmatrix} =
\begin{bmatrix}
\frac{\partial f\_{1}}{\partial c\_{1}} \\
\frac{\partial f\_{2}}{\partial c\_{2}} \Big|\_{\{c\_{1}^{'},c\_{1}^{'}\}} \\
\frac{\partial f\_{2}}{\partial c\_{1}} \Big|\_{\{c\_{1}^{'},c\_{1}^{'}\}} \\
\end{bmatrix} \frac{\partial f\_{2}}{\partial c\_{2}} \Big|\_{\{c\_{1}^{'},c\_{1}^{'}\}}
\begin{bmatrix}
c\_{1}(t) \\
c\_{2}(t)
\end{bmatrix} =
\mathbf{J} \begin{bmatrix}
c\_{1}(t) \\
c\_{2}(t)
\end{bmatrix}
\tag{7}
$$

**J***c c* 1 2 , represents the Jacobian matrix of the linearized dynamical system in equation 6. The eigenvalues 1 2 , of **J***c c* 1 2 , can be calculated from the characteristic equation: **J I** *c c* 1 2 , 0 . Depending on the nature of the eigenvalues, it is possible to know the arrangement of the trajectories near each steady point of the nonlinear system (Figure 6). This linearization process can be extended to perform the phase space analysis of higher dimensional nonlinear dynamical systems (Edelstein-Kesher, 2005).

An important question that arises at this point is whether the behavior of the trajectories obtained from equation (7) accurately reflects the real behavior of the trajectories of the original nonlinear system. Andronov et al. (1973) show that this is the case. If the linearized system has a saddle, a node or a spiral at a given steady point, then the original nonlinear system really has also a saddle, a node or a spiral at that steady point. Furthermore, if a steady point is a stable saddle or node of the linearized dynamical system, then is also a stable saddle or node of the nonlinear system. In this case, the neglected nonlinear terms of equations (5) and (6) practically have not effect on the stability of these points when Re() 0 for both eigenvalues. This kind of steady points is known as **hyperbolic points**, and they are not affected by the small nonlinear perturbations. The topological implication of this fact is that the vector field corresponding to a saddle or a node is not altered by small nonlinear perturbations and, as consequence, has **structural stability** (Strogatz, 1994).

When the eigenvalues of the Jacobian matrix are pure imaginary = *i* , the steady point is a center. The trajectories around this point are closed orbits that are stable. However, the neglected nonlinear terms in equations (5) and (6) can produce an imperfect closure of the orbit, giving rise to a spiral. In this form, the vector field corresponding to a center is altered by small nonlinear perturbations that transform the center into a spiral and, as consequence, has not structural stability (Strogatz, 1994).

According to their stability, steady points of 2-dimensional dynamical systems can be classified into a: 1) **Robust case**, which includes repellers or sources, for which both eigenvalues have Re() > 0; attractors or sinks, for which both eigenvalues have Re() < 0 and saddles, for which one eigenvalues has Re() > 0 and the other one has Re() < 0; 2) **Marginal case**, which includes centers for which both eigenvalues are pure imaginary, and non-isolated steady points for which one eigenvalue has Re() = 0 (Strogatz, 1994).

However, the phase space of a nonlinear system can exhibit another kind of closed orbits called **limit cycles**, which cannot be observed in linear systems. A limit cycle is an isolated

Dynamical Aspects of Apoptosis 255

field **c fc** ( ) that is continuously differentiable on an open set of the plane containing R and 2) R does not contain any fixed point (Edelstein-Kesher, 2005). A consequence of this theorem is that in a 2-dimensional phase space any trajectory trapped into a closed bounded

However, in higher dimensional systems the Poincaré-Bendixon theorem does not longer apply and trajectories can be trapped into a closed region of the phase space without converge into a limit cycle or settle down to a fixed point, and they could be attracted by a complex geometric object called strange attractor, which is a fractal set on which the motion is aperiodic and sensitive to very small changes in initial conditions. This sensitivity makes the motion unpredictable as *t* increases, giving rise to a chaotic dynamics (Strogatz, 1994).

The qualitative features of the vector field of a biochemical dynamical system are strongly dependent on the set of parameters of its corresponding set of differential equations. As the value of one of these parameters changes, the qualitative features of the vector field undergo local variations around the steady points. This parameter-dependent change in the local topological structure of a vector field is known as bifurcation. They generally occur in a onedimensional subspace, and the rest of the dimensions of the phase space are affected as consequence of the flow that can be attracted or repelled from this subspace (Strogatz, 1994; Edelstein-Kesher, 2005). Taking into consideration the imaginary plane, we can roughly classify bifurcations into two cases: 1) the eigenvalues of the Jacobian matrix are both real

bifurcation comprises the saddle-node bifurcation; the transcritical bifurcation, and the subcritical and supercritical pitchfork bifurcation. 2) The eigenvalues of the Jacobian matrix are complex conjugated. Bifurcations occur crossing the imaginary axis as certain parameter changes. This kind of bifurcation comprises the supercritical and subcritical Hopf

In the first case, a) the saddle-node bifurcation causes local variations in the vector field

bifurcation has interesting applications in some models of biological processes that imply the presence of chemical switches; b) a transcritical bifurcation occurs when two steady

of an ordinary differential equation (ODE) that exhibits a subcritical pitchfork bifurcation is

only real steady point *c* = 0 is stable. For > 0 there is an unstable steady point at *c* = 0 and

In the second case, the presence of a Hopf bifurcation leads the system to a limit cycle. As

supercritical Hopf bifurcation drives the transformation of a stable spiral into an unstable spiral that converges to a stable limit cycle (Figure 7). The case of a subcritical Hopf bifurcation is more complicated. A typical example is when an unstable limit cycle shrinks

. When > 0 the only real steady point *c* = 0 becomes unstable. The normal form

varies, when a certain critical value

varies until they collide and annihilate each other. This type of

<0 the steady point *c* = 0 is stable and there are two unstable points at

reaches its critical value

changes. This kind of

changes. These points

varies. c) The normal form

. When < 0 the

*<sup>c</sup>* is reached the

*c*, at which the

and bifurcations occur along the real axis as certain parameter

around two points: a saddle and a node, as a bifurcation parameter

of an ODE that exhibits a supercritical pitchfork bifurcation is <sup>3</sup> *c cc*

points interchange their stability as the bifurcation parameter

.

to cero amplitude as the bifurcation parameter

region R must converge to a limit cycle.

**3.3 Bifurcations**

bifurcation.

<sup>3</sup> *c cc*

*c*

. When

to stable steady points at *c*

the bifurcation parameter

become closer as parameter

Fig. 6. **Classification of the steady points of a 2-dimensional linearized dynamical system. (**a) There is a stable node or attractor in the phase space when both eigenvalues *1* and *<sup>2</sup>* are real and negative (when both eigenvalues are positive, the steady point is a unstable node or repulsor); (b) There is a saddle point in the phase space when both eigenvalues *1* and *<sup>2</sup>* are real, but one of them is positive and the other is negative. The stable manifold is spanned by the eigenvector associated to the negative eigenvalue. The unstable manifold is spanned by the eigenvector associated to the positive eigenvalue. (c) There is a stable spiral in the phase space when both eigenvalues *1* and *<sup>2</sup>* are complex conjugated with negative real part. (d) On the contrary, if both eigenvalues *1* and *<sup>2</sup>* are complex conjugated with positive real part the spiral is unstable. (e) When *1* and *<sup>2</sup>* are pure imaginary the steady point is a center surrounded by a series of stable closed orbits. All figures show the flow of the dynamical system in the phase space spanned by the basis conformed by the variables *c*1*(t)* and *c*2*(t)*. The black point represents the steady point of the dynamical system, and the arrows mark out the direction of the flow of the vector field.

trajectory for which neighbor trajectories can be only spirals that converge to it or diverge from it. If all the spirals converge into the limit cycle, this closed orbit is stable, otherwise is unstable. The existence of this kind of closed trajectories in the plane is settled down by the Poincaré-Bendixson theorem. According to this theorem, exits a trajectory C, which is either a closed orbit or a spiral that converges to a closed orbit as *t* , confined inside a certain closed bounded region R of the plane. This theorem assumes 1) the existence of a vector

Fig. 6. **Classification of the steady points of a 2-dimensional linearized dynamical system.**

real and negative (when both eigenvalues are positive, the steady point is a unstable node or

real, but one of them is positive and the other is negative. The stable manifold is spanned by the eigenvector associated to the negative eigenvalue. The unstable manifold is spanned by the eigenvector associated to the positive eigenvalue. (c) There is a stable spiral in the phase