Mathematical Modeling of the Phoenix Rising Pathway
[['Apoptosis , or programmed cell death , is an important and tightly controlled process in mammalian cells [1] .', 'However , not all cells in the same population undergo apoptosis when exposed to identical death signals [2] , [3] .', 'This “fractional killing” phenomenon is problematic in cancer treatment , but may be beneficial for wound healing since it has been observed that surviving cells in damaged tissues repopulate at a more rapid pace [4] , [5] , [6] .', 'While there could be multiple factors that contribute to the rapid regrowth , one potential mechanism is that apoptotic cells may release signals that can promote proliferation of surrounding cells through the “Phoenix Rising” ( PR ) pathway discovered recently in our lab [5] , [6] .', 'This pathway may play important roles in both regeneration of damaged normal tissues and recurrence of tumors after chemotherapy/radiation therapy .', 'Wound healing in normal tissues is a complicated process that is time-dependent and requires coordination of different cells .', 'While it is known that inflammation is the initial response to tissue damage , the exact cellular and molecular events in wound healing remain unclear .', 'It has been generally assumed that factors released from damaged tissues mobilize and recruit stem and progenitor cells to the damaged site , where they proliferate , differentiate , and eventually replace the damaged tissue [6] , [7] .', 'Our previous studies have shown that two of the key molecular players in the initial response are caspase 3 ( C3 ) and caspase 7 ( C7 ) , which are two proteases activated during the execution phase of apoptosis [5] , [6] .', 'Mice lacking either of these caspases are deficient in skin wound healing and liver regeneration [6] .', 'The activation of C3 and C7 triggers a cascade of molecular events that lead to upregulation of prostaglandin E2 ( PGE2 ) , a growth-promoting signal that stimulates stem and progenitor cell proliferation and thus tissue regeneration .', 'Tumor recurrence often happens after chemotherapy and radiation therapy due to incomplete killing of tumor cells [8] , [9] .', 'Our previous studies have shown that apoptotic cells in the tumor mass can release signals to stimulate proliferation of remaining cells [5] , [6] .', 'Here , C3 in apoptotic cells is again a key regulator for the upregulation of signals that promote tumor regrowth .', 'The PR pathway outlined in our previous studies involves a complicated network of molecular interactions [5] , [6] ( see also Figure 1 ) .', 'To understand the dynamics of these interactions , we developed a mathematical model that links the concentrations of activated C3 , activated C7 , and nuclear factor κB ( NFκB ) to the activity of PGE2 in the PR pathway .', 'This type of “input-output” model , combined with experimental data , has been shown to be useful in understanding mechanisms of molecular events in cells [10] .', 'Our model was built upon previous mathematical models for regulatory networks involved in apoptosis [1] , [11] and arachidonic acid ( AA ) metabolism [12] .', 'Our model was validated by comparing the predicted PGE2 concentrations in MEF and 4T1 cells at 48 hours after 10 Gray ( Gy ) radiation with the experimental data observed in previous studies [5] , [6] .', 'Using this model , we numerically simulated time-dependent changes in levels of key molecular players in the PR pathway after radiation therapy , and evaluated effects of C3 activation , C7 activation , C3 knockout , and cyclooxygenase-2 ( COX2 ) inhibition on PGE2 production .'], ['The PR pathway shown in Figure 1 was modeled as a molecular network with seventeen key interactions .', 'The model development was an iterative process , and involved various simplifications and assumptions ( for details see the Materials and Methods section ) .', 'The final outline of the model is shown in Figure 2 .', 'To validate the model , we compared numerically simulated [PGE2] with experimental data of [PGE2] reported in our previous study for wild-type MEF and 4T1 cells with/without radiation ( see Figure 5b in Ref . [5] ) .', 'It should be noted that two units are used for concentrations reported in this paper: pg mL−1 and µM .', 'The former is used for [PGE2] in order to directly compare it with experimental data in the literature; the latter is used for concentrations of all other molecules in the model .', 'The conversion factor for [PGE2] is 3 . 525×105 pg mL−1 per µM .', 'The maximal difference between model predictions and experimental data of [PGE2] was 9% ( see Table 1 ) , which was smaller than the uncertainties in the experimental data .', 'Using the model and the baseline values of the model constants listed in Tables 2 and 3 , we examined the time-dependent changes in [PGE2] after 10-Gy radiation treatment .', 'The numerical results shown in Figure 3 demonstrated that there was a time delay of ∼24 hours in PGE2 production , due to slow activations of C3 and C7 .', 'After the delay , [PGE2] in wild-type MEF and 4T1 cells as well as C7 knockout ( KO ) MEF cells experienced a short , rapid increase phase , followed by a slow increase phase .', 'In contrast , the simulated production of PGE2 in C3 KO MEF cells was slow and increased linearly with time between 24 and 48 hours .', 'To investigate the dependence of [PGE2] on activation rates of C3 and C7 , we simulated [PGE2] in MEF cells at 48 hours after radiation for a set of values of k1 and k3 .', 'The numerical results shown in Figure 4 demonstrated that [PGE2] was insensitive to changes in k3 when k1 was maintained at its baseline , but increased significantly with increasing k1 when k3 was maintained at its baseline , suggesting that PGE2 production was regulated mainly by the amount of activated C3 ( C3* ) but not activated C7 ( C7* ) .', 'To simulate effects of C3* or C7* deficiency on PGE2 production in irradiated cells , we assumed [C3*] ( or [C7*] ) be zero in C3 ( or C7 ) knockout cells .', 'The numerical results shown in Figure 4 demonstrated that in C3 ( or C7 ) knockout cells , activation of C7 ( or C3 ) was important for increasing the production of PGE2 .', 'Taken together , our model predicted that activation of C3 was critical for PGE2 production whereas activation of C7 was important only in C3 knockout cells .', 'This model prediction was consistent with experimental observations in vivo reported in our previous study [6] , which showed that C3 deficiency could make MEF cells inefficient in promoting stem and progenitor cell proliferation .', 'When both C3 and C7 were knocked out , our mathematical model predicted no PGE2 production in apoptotic cells .', 'This prediction was also consistent with an experimental observation that lethally irradiated MEF cells with double knockout of C3 and C7 were less capable to promote stem or progenitor cell proliferation than those with only C3 knocked out [6] .', 'Nuclear factor κB ( NFκB ) is critical for the regulation of PGE2 production through the control of COX2 expression ( see Figure 2 ) [13] .', 'To simulate effects of NFκB on PGE2 production , we fixed [NFκB] at different levels , and calculated the steady state [PGE2] in unirradiated 4T1 cells .', 'The numerical results are shown in Figure 5; and similar results were also observed for unirradiated MEF cells ( data not shown ) .', 'These profiles demonstrated that the dependence of [PGE2] on [NFκB] variation was biphasic , which was controlled by the rate of COX2 transcription ( see the equation for v10 ) .', 'When [NFκB] was <10−4 µM , the simulated [PGE2] was close to zero .', 'When [NFκB] was increased from 10−2 to 1 µM , a two-order-of-magnitude change , the increase in simulated [PGE2] was minimal .', 'This was because the increase in simulated concentration of COX2 mRNA ( COX2t ) was only 21% , which was too small to cause a significant increase in the simulated [PGE2] .', 'In studies reported in the literature , [NFκB] in untreated cells is often higher than 0 . 1 µM [14] , suggesting that the PR pathway was insensitive to radiation- or other tissue damage-induced increase in [NFκB] .', 'Dynamic changes in numerically simulated concentrations of calcium independent phospholipase A2 ( iPLA2 ) , its active form ( iPLA2* ) , arachidonic acid ( AA ) , and prostaglandin H2 ( PGH2 ) in 4T1 and MEF cells are shown in Figures 6 and 7 .', 'Differences in the corresponding profiles between MEF and 4T1 cells were small .', 'The overall observation was that the simulated concentration of iPLA2* in cells treated with 10-Gy radiation increased with time , which led to an increase in AA production from phospholipids .', 'The numerically simulated [AA] profiles shown in Figures 6 and 7 were quantitatively similar to those observed in previous experimental studies ( see Figure 5a in both Ref . [5] and Ref . [6] ) .', 'To simulate effects of C3 knockout on the PR pathway , we let [C3*] be zero .', 'The simulated concentrations of iPLA2 , iPLA2* , AA , and PGH2 in C3 knockout MEF cells are shown in Figure 8; and the dynamic change in simulated [PGE2] is shown in Figure 3 .', 'These results demonstrated that C3 gene knockout reduced the amount of activated iPLA2 by a factor of 3 . 5 , thereby slowing the production of AA , which in turn reduced the levels of PGH2 and PGE2 .', 'Although the simulated concentrations of PGH2 and PGE2 would continue to increase with time beyond the 48-hour period , the increase may not happen in experimental studies .', 'This is because our mathematical model did not consider cell proliferation nor the inactivation of C7* and NFκB that may happen in cells over time .', 'More importantly , the cells may have already died/lysed after 48 hours .', 'Effects of silencing iPLA2 expression on PGE2 production was simulated by reducing the rate of iPLA2 synthesis ( k−5 ) in wild type MEF cells .', 'The numerical results are shown in Figure 9 , which demonstrated that the simulated [PGE2] at 48 hours post radiation decreased approximately linearly with decreasing k−5 .', 'This result was qualitatively consistent with our previous experimental observation where iPLA2 knockdown with shRNA reduced [PGE2] in MEF cells ( Figure 5b in Ref . [6] ) .', 'Previous studies have shown that exogenous COX2 inhibitors can enhance efficacy of radiation therapy of cancer [15] , [16] , [17] .', 'The mechanism of enhancement is likely to be related to the reduction in the release of growth-promoting signals , such as PGE2 , from apoptotic cells .', 'To simulate this mechanism , we introduced a parameter α to model competitive inhibition of COX2 ( see the Materials and Methods section ) , and numerically simulated the dependence of [PGE2] on α at 48 hours post radiation .', 'The simulation results shown in Figure 10 indicated that [PGE2] decreased rapidly with increasing the value of α , suggesting that inhibition of COX2 could indeed reduce the production of growth-promoting signals in apoptotic cells .', 'We also examined the two negative feedback mechanisms ( Ki14 and Ki16 ) shown in Figure 2 and their roles played in the regulation of PGE2 production in silico .', 'We observed that removal of both negative feedback mechanisms ( i . e . , setting both 1/Ki14 and 1/Ki16 to zero ) led to <2% increase in [PGE2] , suggesting that these inhibitory mechanisms were weak and insignificant in the regulation of tissue regeneration after radiation treatment .', 'Finally , we performed a sensitivity analysis of model predictions .', 'It was observed that [PGE2] was sensitive to only four rate constants: k−5 , k6 , k7 , and k17 , among all 25 rate and equilibrium constants ( see Figure 11 ) .', 'Approximately , [PGE2] was proportional to changes in k−5 and k6 , and reversely proportional to changes in k7 and k17 .', 'In the mathematical model , k17 determined the rate of PGE2 degradation , k6 and k7 determined the rates of AA production and degradation , respectively , and k−5 affected AA production indirectly through regulation of iPLA2 synthesis .', 'Therefore , the analysis suggested that the simulated concentration of PGE2 was sensitive to its rate of degradation and the intracellular concentration of AA .', 'In summary , the mathematical model of the PR pathway yielded new insights into the process of growth-promoting signal production in apoptotic cells .', 'While the model is limited by uncertainty in some parameter values and is only as good as the assumptions that were made , it provides useful information on the PR pathway .', 'The model can be used to integrate experimental data obtained from different studies , adjust for cell-to-cell variability observed in different experiments , and determine sensitivity of the PR pathway to individual molecular interactions .', 'The model can be further improved to serve as a tool for in silico experiments in future studies that may generate experimentally testable hypotheses , and facilitate development of novel strategies for improving cancer treatment and normal tissue regeneration .'], ['To determine [C3*] , we analyzed digital images of Western blot gels reported in our previous study [5] .', 'For each C3 or C3* band in the images , we inversed its intensity and measured the total intensity over the entire area covered by the band , using the Gel Analysis function in ImageJ software .', 'After background intensity subtraction , the total intensity data was used as a measure of the relative protein concentration .', 'To determine the absolute concentration of C3 and C3* , we assumed that the concentration of C3 in unirradiated cells was 200 nM [11] , and then used its band intensity to normalize those of C3* under different experimental conditions .', '[C3*] was determined as the normalized image intensity of C3* multiplied by 200 nM .', 'The baseline values of some model constants listed in Tables 2 and 3 were assumed in this study .', 'Specifically , the cell volume was assumed to be 1 pL , which means that 600 molecules per cell is 1 nM [28] .', 'The degradation rates were assumed to be 0 . 6 min−1 for all mRNAs and 0 . 06 min−1 for all proteins [11] , [21] .', 'The baseline value of α is zero , i . e . , there was no exogenous COX2 inhibitor unless indicated otherwise .', 'k6 was assumed to be 6×103 min−1 .', '[PGES] was assumed to be 0 . 5 µM [12] .', 'The baseline values of other model constants were either obtained directly from the literature or estimated in this study based on model assumptions , experimental data reported in the literature , or values of similar constants used in previous mathematical models .', 'β .', 'This constant depends on experimental conditions .', 'For experiments reported in Ref [5] , 2×105 cells were cultured in 1 mL medium .', 'Thus , the total volume of cells was 2×105 pL; and β was approximately equal to 5000 .']]
['Introduction', 'Results/Discussion', 'Materials and Methods']
['Apoptosis is a tightly controlled process in mammalian cells .', 'It is important for embryogenesis , tissue homoeostasis , and cancer treatment .', 'Apoptosis not only induces cell death , but also leads to the release of signals that promote rapid proliferation of surrounding cells through the Phoenix Rising ( PR ) pathway .', 'To quantitatively understand the kinetics of interactions of different molecules in this pathway , we developed a mathematical model to simulate the effects of various changes in the PR pathway on the secretion of prostaglandin E2 ( PGE2 ) , a key factor for promoting cell proliferation .', 'These changes include activation of caspase 3 ( C3 ) , caspase 7 ( C7 ) , and nuclear factor κB ( NFκB ) .', 'In addition , we simulated the effects of cyclooxygenase-2 ( COX2 ) inhibition and C3 knockout on the level of secreted PGE2 .', 'The model predictions on PGE2 in MEF and 4T1 cells at 48 hours after 10-Gray radiation were quantitatively consistent with the experimental data in the literature .', 'Compared to C7 , the model predicted that C3 activation was more critical for PGE2 production .', 'The model also predicted that PGE2 production could be significantly reduced when COX2 expression was blocked via either NFκB inactivation or treatment of cells with exogenous COX2 inhibitors , which led to a decrease in the rate of conversion from arachidonic acid to prostaglandin H2 in the PR pathway .', 'In conclusion , the mathematical model developed in this study yielded new insights into the process of tissue regrowth stimulated by signals from apoptotic cells .', 'In future studies , the model can be used for experimental data analysis and assisting development of novel strategies/drugs for improving cancer treatment or normal tissue regeneration .']
['Apoptosis , or programmed cell death , is known to be important for embryogenesis , tissue homoeostasis , and cancer treatment .', 'Furthermore , researchers have recently observed that apoptosis may promote wound healing and tissue regeneration , and accelerate undesired solid tumor regrowth after chemotherapy/radiation therapy .', 'Mechanisms of apoptosis-induced tissue regrowth are related to a molecular network discovered recently in our lab .', 'To quantitatively understand the kinetics of interactions of different molecules in this network , we developed a mathematical model and validated it by comparing the simulation results to experimental data reported in previous studies .', 'To gain new insights into the process of tissue regrowth after inducing apoptosis , we used the model to simulate the effects of radiation on the production of a key growth stimulating factor , PGE2 , in apoptotic cells .', 'Additionally , we simulated how PGE2 production could be altered when cells were treated with different inhibitors .', 'We expect that the new mathematical model can be used in future studies to facilitate design of better approaches to cancer treatment or normal tissue regeneration .']
['bioengineering', 'systems biology', 'biomedical engineering', 'biological systems engineering', 'biology', 'computational biology', 'engineering']