18.2: Brownian Motion with Drift and Scaling
Basic Theory
Definition
We start with the assumptions that govern standard Brownian motion, except that we relax the restrictions on the parameters of the normal distribution.
Suppose that \( \mu \in \R \) and \( \sigma \in (0, \infty) \). Brownian motion with drift parameter \( \mu \) and scale parameter \( \sigma \) is a random process \(\bs{X} = \{X_t: t \in [0, \infty)\}\) with state space \(\R\) that satisfies the following properties:
- \( X_0 = 0 \) (with probability 1).
- \( \bs{X} \) has stationary increments. That is, for \( s, \, t \in [0, \infty) \) with \( s \lt t \), the distribution of \( X_t - X_s \) is the same as the distribution of \( X_{t - s} \).
- \( \bs{X} \) has independent increments. That is, for \( t_1, t_2, \ldots, t_n \in [0, \infty) \) with \( t_1 \lt t_2 \lt \cdots \lt t_n \), the random variables \( X_{t_1}, X_{t_2} - X_{t_1}, \ldots, X_{t_n} - X_{t_{n-1}} \) are independent.
- \( X_t \) has the normal distribution with mean \( \mu t \) and variance \( \sigma^2 t \) for \( t \in [0, \infty) \).
- With probability 1, \( t \mapsto X_t \) is continuous on \( [0, \infty) \).
Note that we cannot assign the parameters of the normal distribution of \(X_t\) arbitrarily. We know that since \(\bs{X}\) has stationary, independent increments, \(\E(X_t)\) and \(\var(X_t)\) must be linear functions of \( t \in [0, \infty) \).
Open the simulation of Brownian motion with drift and scaling. Run the simulation in single step mode several times for various values of the parameters. Note the behavior of the sample paths. For selected values of the parameters, run the simulation 1000 times and compare the empirical density function and moments to the true density function and moments.
It's easy to construct Brownian motion with drift and scaling from a standard Brownian motion, so we don't have to worry about the existence question.
Relation to standard Brownian motion.
- Suppose that \(\bs{Z} = \{Z_t: t \in [0, \infty)\}\) is a standard Brownian motion, and that \(\mu \in \R\) and \(\sigma \in (0, \infty)\). Let \(X_t = \mu t + \sigma Z_t\) for \(t \in [0, \infty)\). Then \(\bs{X} = \{X_t: t \in [0, \infty)\}\) is a Brownian motion with drift parameter \(\mu\) and scale parameter \(\sigma\).
- Conversely, suppose that \(\bs{X} = \{X_t: t \in [0, \infty)\}\) is a Brownian motion with drift parameter \(\mu \in \R\) and scale parameter \(\sigma \in (0, \infty)\). Let \(Z_t = (X_t - \mu t) \big/ \sigma\) for \(t \in [0, \infty)\). Then \(\bs{Z} = \{Z_t: t \in [0, \infty)\}\) is a standard Brownian motion.
Proof
It's straightforwrd to show that the processes \(\bs{X}\) and \(\bs{Z}\) satisfy the appropriate set of assumptions.
In differential form, part (a) can be written as \[ d X_t = \mu \, dt + \sigma \, d Z_t, \; X_0 = 0 \]
Finite Dimensional Distributions
Suppose that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is Brownian motion with drift parameter \(\mu \in \R\) and scale parameter \(\sigma \in (0, \infty)\). It follows from part (d) of the definition that \( X_t \) has probability density function \( f_t \) given by \[ f_t(x) = \frac{1}{\sigma \sqrt{2 \pi t}} \exp\left[-\frac{1}{2 \sigma^2 t} (x - \mu t)^2\right], \quad x \in \R \] This family of density functions determines the finite dimensional distributions of \( \bs{X} \).
If \( t_1, t_2, \ldots, t_n \in (0, \infty) \) with \( 0 \lt t_1 \lt t_2 \cdots \lt t_n \) then \( (X_{t_1}, X_{t_2}, \ldots, X_{t_n}) \) has probability density function \( f_{t_1, t_2, \ldots, t_n} \) given by \[ f_{t_1, t_2, \ldots, t_n}(x_1, x_2, \ldots, x_n) = f_{t_1}(x_1) f_{t_2 - t_1}(x_2 - x_1) \cdots f_{t_n - t_{n-1}}(x_n - x_{n-1}), \quad (x_1, x_2, \ldots, x_n) \in \R^n \]
Proof
This follows because \(\bs{X}\) has stationary, independent increments.
\( \bs{X} \) is a Gaussian process with mean function mean function \( m \) and covariance function \( c \) given by
- \( m(t) = \mu t \) for \( t \in [0, \infty) \)
- \( c(s, t) = \sigma^2 \min\{s, t\} \) for \( s, \, t \in [0, \infty) \).
Proof
The fact that \(\bs{X}\) is a Gaussian process follows from the construction \(X_t = \mu t + \sigma Z_t\) for \(t \in [0, \infty)\), where \(\bs{Z}\) is a standard Brownian motion. We know that \(\bs{Z}\) is a Gaussian process. The form of the mean and covariance functions follow because \(\bs{X}\) has stationary, independent increments. Note that \(\mu\) and \(\sigma^2\) are the mean and variance of \(X_1\).
The correlation function is independent of the parameters, and thus is the same as for standard Brownian motion. This is hardly surprising since correlation is a standardized measure of association. \[ \cor(X_s, X_t) \frac{\sigma^2 \min\{s, t\}}{\sigma s \sigma t} = \frac{\min\{s, t\}}{s t} = \sqrt{\frac{\min\{s, t\}}{\max\{s, t\}}}, \quad (s, t) \in [0, \infty)^2 \]
Transformations
There are a couple simple transformations that preserve Brownian motion, but perhaps change the drift and scale parameters. Our starting place is a Brownian motion \( \bs{X} = \{X_t: t \in [0, \infty)\} \) with drift parameter \( \mu \in \R \) and scale parameter \( \sigma \in (0, \infty) \). Our first result involves scaling \( \bs{X} \) is time and space (and possible reflecting in the spatial origin).
Let \( a \in \R \setminus \{0\} \) and \( b \in (0, \infty) \). Define \( Y_t = a X_ {b t} \) for \( t \ge 0 \). Then \( \bs{Y} = \{Y_t: t \ge 0\} \) is also a Brownian motion with drift parameter \( a b \mu \) and scale parameter \(\left|a\right| \sqrt{b} \sigma \).
Proof
Clearly the new process is still a Gaussian process. The mean function is \( \E(Y_t) = a \E(X_{b t}) = a b \mu t \) for \( t \in [0, \infty) \). The covariance function is \( \cov(Y_s, Y_t) = a^2 \cov(X_{bs}, X_{bt}) = a^2 \sigma^2 \min\{b s, b t\} = a^2 b \sigma^2 \min\{s, t\} \) for \( (s, t) \in [0, \infty)^2 \). Finally, since \( \bs{X} \) is continuous, so is \( \bs{Y} \).
Suppose that \( a \gt 0 \) in the previous theorem, so that we are scaling temporally and spatially. In order to preserve the original drift parameter \(\mu\) we must have \(a b = 1\) (if \(\mu \ne 0\)). In order to preserve the original scale parameter \(\sigma\), we must have \( a \sqrt{b} = 1 \). We can't have both unless \(\mu = 0\), which leads to a slight generalization of one of our results for standard Brownian motion:
Suppose that \(\bs{X}\) is a Brownian motion with drift parameter \(\mu = 0\) and scale parameter \(\sigma > 0\). Suppose also that \(c \gt 0 \) and let \(Y_t = \frac{1}{c} X_{c^2 t}\) for \(t \ge 0\). Then \(\bs{Y} = \{Y_t: t \in [0, \infty)\}\) is also a Brownian motion with drift parameter 0 and scale parameter \(\sigma\).
Our next result is related to the Markov property, which we explore in more detail below. We return to the general case where \(\bs{X} = \{X_t: t \in [0, \infty)\}\) is a Brownian motion with drift parameter \(\mu \in \R\) and scale parameter \(\sigma \in (0, \infty)\). If we
restart
Brownian motion at a fixed time \( s \), and shift the origin to \( X_s \), then we have another Brownian motion with the same parameters.
Fix \( s \in [0, \infty) \) and define \( Y_t = X_{s + t} - X_s \) for \( t \ge 0 \). Then \( \bs{Y} = \{Y_t: t \in [0, \infty)\} \) is also a Brownian motion with the same drift and scale parameters.
Proof
Clearly \( \bs{Y} \) is also a Gaussian process. Moreover, \( \E(Y_t) = \E(X_{s + t}) - \E(X_s) = \mu(s + t) - \mu s = \mu t \) for \( t \in [0, \infty) \). Also, if \( r, \, t \in [0, \infty) \) with \( r \le t \) then \begin{align} \cov(Y_r, Y_t) & = \cov(X_{s + r} - X_s, X_{s + t} - X_s) \\ & = \cov(X_{s + r}, X_{s + t}) - \cov(X_{s + r}, X_s) - \cov(X_s, X_{s + t}) + \cov(X_s, X_s) \\ & = \sigma^2 (s + r) - \sigma^2 s - \sigma^2 s + \sigma^2 s = \sigma^2 r \end{align} Finally, \( \bs{Y} \) is continuous by the continuity of \( \bs{X} \).
The Markov Property and Stopping Times
As usual, we start with a Brownian motion \( \bs{X} = \{X_t: t \in [0, \infty)\} \) with drift parameter \( \mu \) and scale parameter \( \sigma \). Recall again that a Markov process has the property that the future is independent of the past, given the present state. Because of the stationary, independent increments property, Brownian motion has the property. As a minor note, to view \( \bs{X} \) as a Markov process, we sometimes need to relax Assumption 1 and let \( X_0 \) have an arbitrary value in \( \R \). Let \( \mathscr{F}_t = \sigma\{X_s: 0 \le s \le t\} \), the sigma-algebra generated by the process up to time \( t \in [0, \infty) \). The family of \(\sigma\)-algebras \(\mathfrak{F} = \{\mathscr{F}_t: t \in [0, \infty)\}\) is known as a filtration.
Brownian motion is a time-homogeneous Markov process with transition probability density \( p \) given by \[ p_t(x, y) = f_t(y - x) =\frac{1}{\sigma \sqrt{2 \pi t}} \exp\left[-\frac{1}{2 \sigma^2 t} (y - x - \mu t)^2\right], \quad t \in (0, \infty); \; x, \, y \in \R \]
Proof
Fix \( s \in [0, \infty) \). The theorem follows from the fact that the process \( \{X_{s+t} - X_s: t \in [0, \infty)\} \) is another standard Brownian motion, as noted above , and is independent of \( \mathscr{F}_s \).
The transtion density \( p \) satisfies the following diffusion equations . The first is known as the forward equation and the second as the backward equation .
\begin{align} \frac{\partial}{\partial t} p_t(x, y) & = -\mu \frac{\partial}{\partial y} p_t(x, y) + \frac{1}{2} \sigma^2 \frac{\partial^2}{ \partial y^2} p_t(x, y) \\ \frac{\partial}{\partial t} p_t(x, y) & = \mu \frac{\partial}{\partial x} p_t(x, y) + \frac{1}{2} \sigma^2 \frac{\partial^2}{ \partial x^2} p_t(x, y) \end{align}Proof
These results follows from standard calculus.
The diffusion equations are so named, because the spatial derivative in the first equation is with respect to \( y \), the state forward at time \( t \), while the spatial derivative in the second equation is with respect to \( x \), the state backward at time 0.
Recall again that a random time \( \tau \) taking values in \( [0, \infty] \) is a stopping time with respect to the process \( \bs{X} \) if \( \{\tau \le t\} \in \mathscr{F}_t \) for every \( t \in [0, \infty) \). The \( \sigma \)-algebra associated with \( \tau \) is \[ \mathscr{F}_\tau = \left\{B \in \mathscr{F}: B \cap \{\tau \le t\} \in \mathscr{F}_t \text{ for all } t \ge 0\right\} \] See the section on Filtrations and Stopping Times for more information on filtrations, stopping times, and the \(\sigma\)-algebra associated with a stopping time. Brownian motion \( \bs{X} \) is also a strong Markov process.
Suppose that \( \tau \) is a stopping time and define \( Y_t = X_{\tau + t} - X_\tau \) for \( t \in [0, \infty) \). Then \( \bs{Y} = \{Y_t: t \in [0, \infty)\} \) is a Brownian motion with the same drift and scale parameters, and is independent of \( \mathscr{F}_\tau \).


16.2: Responses to Pain
By the end of this section, you will be able to:
- Describe the physiologic response to pain
- Identify behavioral response to pain
- Identify affective response to pain
Patients can have many different responses to pain. The body perceives pain as a sign of danger and will immediately initiate responses to help the person stay safe. The sympathetic and parasympathetic nervous system s play a big role in physiologic responses to pain. Many patients may experience sympathetic body changes to a perceived threat of pain. Behavioral and emotional responses can also be observed in patients experiencing pain. Vocalization, facial expressions, movement, and negative emotions can all be responses to pain.
Patients may experience a wide variety of behaviors when in pain. Patients may lash out or become quiet and withdrawn when in pain. Patients in acute pain are more likely to vocalize their pain and exhibit facial expressions of pain. Patients suffering from chronic pain may struggle with anxiety and depression from the continued feeling of pain. Patients may even suffer from an extreme fear of pain and avoid any situation that may cause pain.
Physiologic Response
Pain warns a person of danger to prevent damage to the body (Arcuri, 2022). The physiologic response to pain is the body’s involuntary responses to a painful stimulus such as inflammation or changes in heart rate and blood pressure. Physiologic responses can include sympathetic nervous system reactions and behavioral reactions such as facial expressions, emotions, vocalizations, and movement. Many of the observed nonverbal cues from patients are physiologic responses to pain.
Definition: Recognize the patient or designee as the source of control and full partner in providing compassionate and coordinated care based on respect for patient’s preferences, values, and needs.
Knowledge: Demonstrate comprehensive understanding of the concepts of pain and suffering, including physiologic models of pain and comfort.
Skill: Assess presence and extent of pain and suffering. The nurse will:
- Focus on the duration, location, and etiology of the patient’s pain.
- Ask the patient how long the pain has lasted to determine if the pain is acute, chronic, or breakthrough pain.
- Ask the patient to identify the location of the pain.
- Ask the patient to describe how the pain feels and what causes or makes the pain worse to try to determine the cause of pain.
Attitude: Appreciate the role of the nurse in relief of all types and sources of pain or suffering.
Sympathetic Nervous System
The autonomic nervous system contains the sympathetic and parasympathetic nervous systems and regulates the involuntary physiologic responses of the body (Figure 16.5). The role of the sympathetic nervous system is to respond to perceived stressful or dangerous situations. The main response from the sympathetic nervous system is the fight-or-flight response , which triggers the body to either run away from or fight a perceived threat. For example, the body may perceive an encounter with a wild animal as a threat. The sympathetic nervous system will activate and tell the person to run away from the animal to safety. The fight-or-flight response may not be activated in perceived less stressful situations, but the body may still illicit a sympathetic response (Yeater et al., 2021). For example, a patient may be experiencing postsurgical pain even with pain medication. The sympathetic nervous system may respond to the pain by elevating the heart rate and blood pressure.
The sympathetic pain response involves many automatic bodily functions such as:
- enlarging pupils to improve vision;
- slowing digestion to allow the body to use energy in other places; and
- increasing heart rate and blood pressure to improve circulation (Cleveland Clinic, 2022b).
Nurses may notice these nonverbal signs of pain in a patient even before the patient states they are in pain. Noticing sympathetic pain responses is especially important in patients who cannot communicate their pain. Patients who are unconscious, confused, nonverbal, young, or cognitively impaired may not be able to adequately express their pain.
Parasympathetic Nervous System
The parasympathetic nervous system opposes the sympathetic nervous system and regulates the automatic bodily functions in times of rest and relaxation. Once the perceived stress or threat is gone, the parasympathetic nervous system takes over and returns the body to its normal functioning (Cleveland Clinic, 2022a).
The parasympathetic response includes:
- constricting pupils to limit light;
- producing saliva and mucus to assist in digestion and breathing;
- lowering heart rate and blood pressure to reduce workload;
- directing energy toward digestion to help break down food;
- directing the pancreas to create insulin to regulate blood sugar; and
- relaxing muscles to assist with urination and defecation (Cleveland Clinic, 2022a).
These responses indicate to the nurse that the patient is experiencing adequate pain relief. In the same way that the nurse observes sympathetic responses to detect a patient’s pain, the nurse may observe parasympathetic responses to determine if pain interventions are effective (Figure 16.5).
Vital signs can be an indicator of pain, and nurses can use vital sign measurements along with the patient’s response to determine if pain interventions are effective. Measuring vital signs as an indicator of pain can be especially useful for patients who cannot verbalize their pain. Pain can cause an increase in blood pressure, heart rate, and respiratory rate. Abnormal vital signs can occur without pain, so it is important that nurses perform other methods of assessment for pain as well. Nurses can also measure vital signs to determine if they have normalized after pain interventions. For example, if a patient has elevated blood pressure and is exhibiting signs of pain, the nurse can assess their blood pressure after pain medication has been given to ensure that it has normalized. This can help indicate that the pain interventions are effective.
Behavioral Response
Behavior plays a big role in pain response. Nurses can easily observe behavioral responses to pain and can measure the severity of pain based on a patient’s behavior. Noticing behavioral pain responses is especially important in patients who cannot communicate their pain.
Behavioral pain responses may include:
- facial expressions,
- vocalization,
- movement,
- emotions, and
- interactions with others.
Behavioral pain responses are often big reactions that others easily notice. For example, when a professional athlete gets injured during a game, everyone in the crowd understands the athlete is in pain due to their behavior. The athlete may fall to the ground, grab the area of their injury, and yell out in pain (Figure 16.6).
Vocalization
Patients will sometimes use sounds, noises, and words to express pain. This is called vocalization ; and it most often occurs in acute pain of varying severity. Vocalization of pain includes crying, screaming, moaning, gasping, or grunting (Helmer et al., 2020). Patients can vocalize their pain by using pain rating scales. Asking patients to rate their pain on a numeric scale can be an effective way to assess patients for pain. Patients can also vocalize pain descriptors such as severity, duration, location, and type of pain.
The use of vocalization to express pain is used often in younger children (Helmer et al., 2020). Younger children have limited communication when trying to express pain. The most common way for infants to express discomfort is by crying. Older children may be capable of stating their pain using appropriate pain scales (Figure 16.7), such as the Wong-Baker FACES Pain Rating Scale , but may still use crying or yelling to express pain (Venable, 2018).
Assessing pain in pediatric patients can be difficult, especially in infants. Nurses must be aware of behavioral responses to pain when caring for infants. Infants primarily express their needs through crying, but not all crying equates to pain. The FLACC (Face, Legs, Activity, Cry, Consolability) scale can be used to assess an infant’s pain. The FLACC scale is recommended to be used in younger patients as it measures appropriate behavioral responses in young children. The FLACC scale scores patients’ pain based on five criteria: facial expressions, body posture, activity, crying, and the ability to console (Trottier et al., 2022). The total score is 0 to 12, with each category receiving 0 to 2 points based on patient response. Nurses can try comfort measures such as swaddling, feeding, and rocking to ease the infant’s crying. If no comfort measures work, the crying may be due to pain (Helmer et al., 2020).
Facial Expression
Facial expressions not only demonstrate emotions such as happiness, fear, or sadness, they can also demonstrate pain. Frowning, closing eyes, clenching teeth, opening the mouth, biting lips, grimacing, furrowing eyebrows, and scowling can all be used to express pain (Figure 16.8) (Cho & Hong, 2020).
Some research shows that social context can change the way a person expresses pain (Kappesser, 2019). For example, a young child may cry and use facial expressions to demonstrate pain around their parents. However, they immediately stop crying and try to smile when their older sibling comes into the room because they want to appear more mature. Another example is a person feeling comfortable enough to show facial expressions of pain in the presence of their partner rather than in front of a colleague or stranger (Kappesser, 2019). Patients may not feel comfortable showing facial expressions when in pain. They may try to hide their facial expressions in the presence of healthcare providers. It is important that nurses always help patients feel comfortable expressing their pain.
Body Movement
Patients may demonstrate pain through movement such as guarding, touching the location of the pain, withdrawing from touch, or tremors (Cho & Hong, 2020). Nurses may notice muscle tension and resistance to examination of the painful location.
Children often show pain through withdrawal of touch. While they may seek the comfort of their parents, they will often resist someone touching or manipulating the painful area due to fear of touch causing more pain. This can be difficult for nurses to assess the location of pain. Severe pain may require pharmacological interventions prior to assessment of the location of pain. Nurses must be mindful of the location of pain when doing a physical assessment. Visual assessment, diagnostic imaging, and descriptors of pain from the patient can help provide a comprehensive pain assessment without touching the painful area.
Social Interaction
The way that patients interact with others can change when they are in pain. Patients in chronic pain may withdraw from social interactions and become isolated. This can make the pain worse and contribute to deteriorating physical health in patients (Bannon et al., 2021). When patients are in pain and around others, they may appear quiet, reserved, and uninterested (Oommen & Shetty, 2020). Nurses can educate patients that these social interactions are a normal response to pain and help them verbalize to their loved ones that these negative emotions are responses to pain. It is important that patients are able to verbalize their pain in social situations, so they do not become socially isolated and withdrawn. Research has shown that supportive social environments can help patients cope with pain (Bannon et al., 2021).
Affective Response
The affective response to pain refers to the emotional interpretation of pain, such as how uncomfortable the pain is. The affective response can help determine how much pain is felt (Cleveland Clinic, 2022a). Negative emotions such as fear and anxiety can be connected to pain. Because pain is a subjective experience, patients may have different emotional responses to pain. For example, an athlete with a knee injury may feel fear along with pain because they are worried the pain means a season-ending injury. Another patient feeling pain from a broken leg may feel anxiety along with the pain because they will not be able to work while recovering from their injury.
Interventions such as cognitive behavioral therapy, guided imagery , and meditation can help patients adjust their affective response to pain because they promote relaxation and decreased stress (Talbot et al., 2019). These interventions can help decrease negative emotions such as fear and anxiety while also decreasing the perception of pain .
Negative Emotion
Pain is an unpleasant sensation and associated with negative emotions. Negative emotions can include anger, frustration, fear, anxiety, depression, guilt, and sadness. Negative emotions can make it difficult to complete daily activities and can lead to social withdrawal. Everyone can feel negative emotions and they are not always connected to pain. However, it is important for patients to understand that pain can bring on negative emotions, especially anxiety and depression. If patients are not able to cope with negative emotions in healthy ways, it can lead to unhealthy consequences (Scott, 2022).
Accepting emotions and finding healthy coping mechanisms is essential to reducing pain and preventing negative consequences. Healthy coping mechanisms for negative emotions can include acceptance and understanding of the emotions, exercise, meditation, and lifestyle changes (Scott, 2022).
Anxiety
Anxiety often goes hand in hand with pain and is more commonly seen in patients with chronic pain. Pain can affect patients’ daily lives and can have a big impact on their physical, mental, and social health. Many patients who experience anxiety with pain may also experience depression. Research shows that 65 percent of patients seeking help for depression also report pain symptoms (Harvard Health Publishing, 2021).
Anxiety and depression can make it harder to treat pain. Patients experiencing anxiety and depression may be less motivated to find ways to treat their pain and may be unable to focus on pain management strategies. Cognitive behavioral therapy, mindfulness, and medications can help treat anxiety, depression, and pain in patients. These therapies can be multimodal in that they can treat the negative emotions and the pain at the same time.
Fear
The perception of pain can be influenced by fear (Oommen & Shetty, 2020). Some patients may even have a phobia of pain. When that phobia of pain is extreme, it’s called algophobia . Patients may experience anxiety and fear at the thought of pain. Algophobia can make patients more sensitive to pain and they may perceive normal activities as a potential cause of pain (Cleveland Clinic, 2021).
Fear of pain can cause patients to catastrophize pain. The act of catastrophizing is the picturing the worst possible outcome for a situation (Cleveland Clinic, 2021). Patients may be intensely focused on avoiding pain and can see the potential of pain in any situation. Patients may become anxious and withdrawn as they try to avoid situations that could cause pain. If patients are experiencing pain, they may avoid situations that they think can make their pain worse. Cognitive behavioral therapy and mindfulness can help patients deal with their fear of pain. Exposure to potential painful situations can also help patients rationalize their fear of pain and continue living their daily lives.


3.16: Retailers and Non-store Channel Types
- Match a retailer description with its non-store channel type
Non-store retailing is the selling of goods and services outside the confines of a retail store, off the premises of fixed retail locations. This distribution channel can be divided into direct selling and distance selling.
Modern direct selling includes sales made through one-on-one demonstrations, hosted product sales events in-home, and other personal contact. Direct selling consists of two main business models: single-level marketing and multi-level marketing.
In single-level marketing, a direct seller buys products from a parent organization and sells them directly to customers. In multi-level marketing, also known as network marketing and person-to-person marketing, the direct seller may earn money from both direct sales to customers and by sponsoring new direct sellers, potentially earning commissions from their efforts. Amway, Avon and Tupperware engage in this form of multi-level marketing.
Distance selling includes:
- Mail Order
- Catalogue Sales
- Telephone Solicitations
- Automated Vending
- Electronic Commerce (e-commerce)
Non-store retailing, sometimes labeled ‘home shopping’, is a relatively small portion of the total retail activity in the United States. However, it’s growing significantly, led by the expansion of e-commerce.
According to the Census Bureau of the U.S Department of Commerce, e-commerce sales accounted for 8.9% of total retail sales in 2017. Estimated at $453.5 billion, they show a 16.0% increase over 2016 levels. Perhaps even more telling is that total retail sales grew at only 4.4%. This means e-commerce is growing 3.6 times faster than other retailing in the United States.
Since 2010 and continuing through 2017, e-commerce sales have grown at around an 11.7% compound annual growth rate (CAGR). It is in this growth rate that we see the changes in consumer behavior we’ve discussed throughout these sections reflected. Namely, connected consumers are using technology to engage with brands and retailers. And, in doing so, they’re blurring channels, without changing expectations for accessibility, consistency or service.
Contributors and Attributions
- Retailers and Non-store Channel Types. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution


4.16: Tissues Worksheet Answers
Circle the correct answer:
1. A tissue is: a collection of cells
2. The four major tissue types are: epithelial / connective / muscle / nervous
3. Epidermal tissue: covers body surfaces / lines hollow organs / forms glands.
4. The inside of blood vessels is lined with: cuboidal epithelium / squamous epithelium / transitional epithelium / columnar epithelium?
5. The tissue illustrated below is: squamous epithelium
6. The kidney tubules are lined with: cuboidal epithelium
7. Where is ciliated columnar epithelium NOT found : Lining the kidney tubule / lining the small intestine? Note: the columnar epithelium lining the gut has microvilli on the surface. The difference between cilia and microvilli can be seen here.
8. Stratified epithelia: consist of several layers of cells
9. Transitional epithelium allows the: bladder to stretch .
10. The tissue illustrated below is: squamous epithelium
11. Non keratinised stratified squamous epithelium does NOT OCCUR: covering the body .
12. The surface cells of keratinised stratified squamous epithelium are continually: being shed from the surface.
13. Which of the following are NOT found in connective tissue?: cilia.
14. Which of the following are NOT connective tissues: fingernails / saliva
15. Which of the following muscle tissues has branched fibres, intercalated discs between adjacent cells and contracts automatically: cardiac muscle
16. Which of the following muscle tissues has long fibres with cross bands and nuclei on the surface? Striated voluntary muscle
17. Which muscle tissue moves bones and facial skin? striated voluntary muscle
18. The tissue illustrated below is: cartilage
19. Which muscle tissue is found in blood vessel walls, in the gut wall and in glands? smooth involuntary muscle
20. The tissue illustrated below is: cardiac muscle
21. Match the tissue type with the function.
| Tissue type | Function |
|---|---|
| a. Connective Tissue | B. Supporting & transporting |
| b. Muscle Tissue | C. Contracting |
| c. Epithelial Tissue | D. Covering and lining |
| d. Nervous Tissue | A. Communicating |


23.12.1: Review Questions
Review Questions
How does a white dwarf differ from a neutron star? How does each form? What keeps each from collapsing under its own weight?
2.
Describe the evolution of a star with a mass like that of the Sun, from the main-sequence phase of its evolution until it becomes a white dwarf.
3.
Describe the evolution of a massive star (say, 20 times the mass of the Sun) up to the point at which it becomes a supernova. How does the evolution of a massive star differ from that of the Sun? Why?
4.
How do the two types of supernovae discussed in this chapter differ? What kind of star gives rise to each type?
5.
A star begins its life with a mass of 5 M Sun but ends its life as a white dwarf with a mass of 0.8 M Sun . List the stages in the star’s life during which it most likely lost some of the mass it started with. How did mass loss occur in each stage?
6.
If the formation of a neutron star leads to a supernova explosion, explain why only three of the hundreds of known pulsars are found in supernova remnants.
7.
How can the Crab Nebula shine with the energy of something like 100,000 Suns when the star that formed the nebula exploded almost 1000 years ago? Who “pays the bills” for much of the radiation we see coming from the nebula?
8.
How is a nova different from a type Ia supernova? How does it differ from a type II supernova?
9.
Apart from the masses, how are binary systems with a neutron star different from binary systems with a white dwarf?
10.
What observations from SN 1987A helped confirm theories about supernovae?
11.
Describe the evolution of a white dwarf over time, in particular how the luminosity, temperature, and radius change.
12.
Describe the evolution of a pulsar over time, in particular how the rotation and pulse signal changes over time.
13.
How would a white dwarf that formed from a star that had an initial mass of 1 M Sun be different from a white dwarf that formed from a star that had an initial mass of 9 M Sun ?
14.
What do astronomers think are the causes of longer-duration gamma-ray bursts and shorter-duration gamma-ray bursts?
15.
How did astronomers finally solve the mystery of what gamma-ray bursts were? What instruments were required to find the solution?


14.2: Exercise
The Phases of Mitosis
Exercise 1: Modeling the Phases of Mitosis with Pop Beads
Materials:
- Chalk (can be used directly on lab bench to draw cellular structures and then washed off)
-
Chromosome modeling kits
- Commercially available pop bead kits (e.g Carolina Biological Supply Company, Item #171100)
- 40 pop beads of one color (red)
- 40 pop beads of another color (yellow)
- 8 magnetic centromeres
- 4 laminated pictures of centrosomes (each consisting of a pair of centrioles)
- Paper towels / Kimwipes
Procedure:
Using models is a great way to represent natural structures and processes that are too small, or too large, or too complex to observe directly. By building chromosomes from the pop beads and manipulating them to model cell division (mitosis and meiosis) you will enhance your understanding of the nature of chromosomes and the cellular structures needed to perform cell division. In this simulation, your cell will be a diploid (2n) cell with 4 chromosomes. It will contain 2 homologous pairs of chromosomes. In order to differentiate between the pairs of homologs, one pair of homologous chromosomes will be longer than the other. One chromosome of each pair is red and represents maternal DNA (genetic material contributed by a female’s egg). The other chromosome of each pair is yellow and represents paternal DNA (genetic material contributed by a father’s sperm). Thus, for each pair of homologous chromosomes, one should be red and one should be yellow. The strands of pop beads represent the DNA in the form of chromatin during the G1, S, and G2 phases. They then represent chromosomes as they enter the phases of mitosis.
This diploid cell with 2 homologous pairs of chromosomes will be modeled as it moves through the following phases of mitosis:
- Interphase (uncondensed DNA) Before Synthesis of DNA (G1)
- Interphase (uncondensed DNA) After Synthesis of DNA (G2)
- Prophase
- Metaphase
- Anaphase
- Telophase
- Cytokinesis
- You will build 2 pairs of homologous chromosomes (2n = 4). The first homologous pair of single chromosomes should be constructed using 12 red beads for one member of the long pair and 12 yellow beads for the other member of the pair. You may place the magnetic centromere at any position along the chromosome, but it must be at the same position on both members of the homologous pair.
- The second homologous pair of single chromosomes should be constructed using 6 red beads for one member of the short pair and 6 yellow beads for the other member of the pair. You may place the magnetic centromere at any position along the chromosome, but it must be at the same position on both members of the homologous pair.
- Use the pop beads to model the G1 phase of interphase (before synthesis of the DNA). On your lab bench, use the chalk to draw the cell membrane, nucleus, and nucleolus. Place a laminated centrosome in the cytoplasm of your cell. Place the 4 assembled chromosomes in the nucleus of your cell drawing. Remember that these represent a mass of chromatin.
- DNA replication takes place during the S phase of interphase. Model the S phase of interphase by assembling a second strand that is identical to each of the 4 single chromosomes. The identical strands are considered sister chromatids, which are held together by their magnetic centromeres. (Note: In a living cell, the centromere is a single unit until it separates in anaphase. Therefore, consider the pair of magnets to be a single centromere.)
- Model the G2 phase of interphase by letting your 4 assembled replicated chromosomes rest in the nucleus of your cell. Duplicate the centrosomes by placing another laminated centrosome in the cytoplasm of your cell drawing. Begin to move them towards opposite poles of the cell drawing.
- To model the prophase stage of mitosis, leave the chromosomes where they are in your cell drawing. In this stage, the chromatin coils and condenses into chromosomes. Using the paper towel (or Kimwipe) start erasing some of the nuclear membrane on your cell drawing. This will simulate the breakdown of the nuclear membrane. Also erase the nucleolus, as the nucleolus disappears in this stage. Continue to move your centrosomes to opposite sides of the cell. Use the chalk to draw spindle fibers beginning to form and radiating outward toward the chromosomes.
- At prometaphase the centrosomes are at opposite poles of the cell. During this phase the chromosomes continue to condense, the nuclear membrane completely breaks down, and the spindle fibers start to attach to the kinetochores of the centromeres. These are called kinetochore microtubules . Other spindle fibers radiate outward from the centrosomes, but these do not attach to the kinetochores. These spindle fibers are called nonkinetochore microtubules.
- To model metaphase , move the centromeres of your chromosomes to lie on an imaginary plane midway between the centrosomes that are now positioned at opposite poles of the cell. Line up the individual chromosomes in the middle (equator) of the cell. Sister chromatids remain attached at the centromere during metaphase.
- To model anaphase , pull the magnetic centromeres apart and slide them towards opposite poles of the cell. Keep them attached to your drawn kinetochore microtubules, but use the paper towel to erase them, making them shorter. With the chalk, lengthen the nonkinetochore microtubules and have them overlap in the middle of your cell. After separation at the centromere, each sister chromatid is now referred to as an individual chromosome. Anaphase ends when the chromosomes reach the opposite ends of the cell.
- Model telophase by piling up your chromosomes at each pole. Erase your spindle fibers, as they disappear during this stage. Even though we can’t simulate it, the chromosomes will start to decondense and uncoil back into chromatin. Redraw a nuclear envelope around each pile of chromosomes. Also add nucleoli to your drawing. The formation of separate nuclear envelopes divide the nuclei and marks the end of telophase.
- The division of the cytoplasm, or cytokinesis, results in the formation of two separate cells. To model cytokinesis in animal cells, fungi, and slime molds, leave the piles of chromosomes at their separate poles. Draw indentations of the cell membrane inwards, towards the cytoplasm, on the sides of the cell where there are no centrosomes. This represents a cleavage furrow ,which eventually “pinches” the cell’s cytoplasm into two separate cells. The proteins actin and myosin contribute to the formation of the cleavage furrow. In plant cells, membrane-bound vesicles migrate to the center of the cell (the equatorial plane) and fuse together to form a cell plate . The cell plate eventually divides the cytoplasm into two separate cells. Materials needed to build the cell wall are released from the vesicles at the cell plate, forming a new cell wall.
- How do the daughter cells you formed compare to the original parent cell? _________________________________________________________________
Observe the phases of Mitosis in Plant Cells
Exercise 2: Observing the Phases of Mitosis in the Onion Root Tip
Materials:
- Prepared slide of the onion root tip
- Compound light microscope
Procedure:
- Examine a slide of a longitudinal section of an onion root tip. Adjust the slide to view the region just above the root cap, where there are likely to be dividing cells.
- Focus on the dividing cells using the 4x scanning objective lens, then switch to the 10x objective and then the 40x objective.
- Survey the slide to find a cell in each phase of mitosis. Draw a cell for each phase in the boxes provided.
Interphase
DNA is uncondensed and in the form of chromatin. Individual chromosomes are not visible. The nuclear membrane is intact. The nucleolus is visible.
Figure \(\PageIndex{1}\): Interphase in plant cells.
Prophase
Chromatin begins to condense into distinguishable chromosomes. These “puffy” structures are seen throughout the nucleus. Nucleoli begin to disappear. In late prophase (often called prometaphase) the nuclear membrane is no longer visible.
Figure \(\PageIndex{2}\): Prophase in plant cells.
Metaphase
The chromosomes line up in the middle of the cell. Spindle fibers attach to kinetochores at the centromere and extend to the poles of the cell.
Figure \(\PageIndex{3}\): Metaphase in plant cells.
Anaphase
Centromeres split, separating each former chromatid into two individual chromosomes. The chromosomes move toward opposite poles.
Figure \(\PageIndex{4}\): Anaphase in plant cells.
Telophase and Cytokinesis
Chromosomes reach the poles. The nuclear envelopes begin to reform. The formation of a cell plate forms between the two cells to carry out cytokinesis.
Figure \(\PageIndex{5}\): Telophase in plant cells.
Figure \(\PageIndex{6}\): Cytokinesis in plant cells.
Observe the phases of Mitosis in Animal Cells
Exercise 3: Observing the Phases of Mitosis in the Whitefish Blastula
Materials:
- Prepared slide of whitefish blastula
- Compound light microscope
Procedure:
The blastula is an early embryonic stage where many of the cells are dividing at any one time.
- Focus on the dividing cells using the 4x scanning objective lens, then switch to the 10x objective and then the 40x objective.
- Survey the slide to find a cell in each phase of mitosis. Draw a cell for each phase in the boxes provided.
Interphase
The DNA is uncondensed and in the form of chromatin. Individual chromosomes are not visible. The nuclear membrane is intact. The nucleolus is visible.
Figure \(\PageIndex{7}\): Interphase in animal cells.
Prophase
Chromatin begins to condense and chromosomes are distinguishable. These “puffy” structures are seen throughout the nucleus. The nucleoli begin to disappear. In late prophase (often called prometaphase) the nuclear membrane is no longer visible.
Figure \(\PageIndex{8}\): Prophase in animal cells.
Metaphase
The chromosomes line up in the middle of the cell. Spindle fibers attach to kinetochores at the centromere and extend to the poles of the cell.
Figure \(\PageIndex{9}\): Metaphase in animal cells.
Anaphase
Centromeres split, separating each former chromatid into two individual chromosomes. The chromosomes move toward opposite poles.
Figure \(\PageIndex{10}\): Anaphase in animal cells.
Telophase and Cytokinesis
Chromosomes reach the poles. The nuclear envelopes begin to reform. A cleavage furrow forms between the two cells to carry out cytokinesis.
Figure \(\PageIndex{11}\): Telophase in animal cells.
Figure \(\PageIndex{12}\): Cytokinesis in animal cells.
- Why would the method of cytokinesis in animal cells not work in plant cells? Explain. __________________________________________________________________
How Long Does a Cell Spend in Each Phase of the Cell Cycle?
Exercise 4: Determining Time Spent in Different Phases of the Cell Cycle (Optional)
Materials:
- Laptop
- Calculator
Procedure:
- Obtain a laptop.
- Open a web browser and go to the following site: http://www.biology.arizona.edu(opens in new window)
- This site will provide an interactive test of your ability to identify the stages of mitosis. It will also allow you to calculate the duration of the stages identified in the laboratory exercise you just completed, but the website will give standard results for the entire class.
- Go to the Cell Biology section, and find the activity “ Online Onion Root Tips ”.
- Keep clicking on “Next” at the bottom of the page until you get to the screen: Determining Time Spent in Different Phases of the Cell Cycle
- Click on “Next” at the bottom of the page.
- Identify each stage shown to you by the program. When a picture of a cell pops up in a stage of mitosis, simply click on the phase in which the cell belongs. If you make a mistake, read the explanation for why you were mistaken before making a new selection.
- When you are finished, use the formula given below and record your results in the table.
- The duration of each stage of mitosis can be determined by using the following formula. Compute the length of time for each stage and place your calculations in the table below.
(Number of cells in a stage ÷ Total number of cells) x 1440 (min in a day) = minutes a cell spends
in each stage in one day
|
Number of cells in each stage |
Interphase ______ |
Prophase ______ |
Metaphase ______ |
Anaphase _______ |
Telophase _______ |
Total 36 |
|---|---|---|---|---|---|---|
|
Percent of cells (# of cells / Total) |
100% |
|||||
|
Time (in minutes) spent in Stage – use calculation above |
References
Belwood, Jacqueline; Rogers, Brandy; and Christian, Jason, Foundations of Biology Lab Manual (Georgia Highlands College). “Lab 10: Mitosis & Meiosis,” (2019). Biological Sciences Open Textbooks . 18. CC-BY
https://oer.galileo.usg.edu/biology-textbooks/18 (opens in new window)


11.13: Exercise- Referencing with a Script
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In this exercise, we’ll add referencing to the script from the previous exercise so that the new script is more useful. You’ll also learn some important strategies along the way.
As you may recall, the ERP CORE data were recorded without a reference (which is possible in only a few recording systems), so we need to use EEG Channel Operations to reference the data. We’ll also add bipolar VEOG and HEOG channels. The N170 is traditionally referenced to the average of all sites (see Chapter 5), and that’s what we’ll do in this example.
You should already have 9 datasets loaded from the previous exercise. If you don’t, run Script4.m again. With the 10_N170 dataset active, select EEGLAB > ERPLAB > EEG Channel operations . Clear out any equations that are already set, set the Mode to Create new dataset , and make sure the box labeled Try to preserve channel locations is checked. Click the Reference Assistant button, and enter the parameters shown in Screenshot 11.5.
The key parameter is avgchan(1:33) , which tells it to use the average of all 33 channels as the reference. Then click OK , which should fill the equations window with an equation for referencing each of the 33 channels to the average of Channels 1–33. Then add these two equations to create bipolar HEOG and VEOG channels:
nch34 = ch31 - ch32 Label HEOG-bipolar
nch35 = ch33 - ch16 Label VEOG-bipolar
Then click Run to execute the routine. Now type eegh to see the script command for EEG Channel Operations, which begins with EEG = pop_eegchanoperator . This command is extremely long, because it contains all 35 equations in it. There’s a better way to do this: Instead of having this huge command, we can put all the equations in a file and pass the filename instead of the list of equations.
Seeing a List of the Channels in a Dataset or ERPset
When you’re using the EEGLAB/ERPLAB GUI, it’s usually pretty easy to see what channels are in a dataset or ERPset. However, this information isn’t so obvious when you’re writing scripts. Here’s a little trick that I use quite often: If you double-click on the name of a variable in the Workspace pain of the main Matlab window, a Variables pane will appear and show the contents of that variable. For example, you can double-click on the EEG variable to see its contents. EEG is a complex structure with many fields, and you can double-click on the name of a field to see the contents of that field. The screenshot below shows what I see when I first double-click on the EEG variable and then double-click on the chanlocs (channel locations) field. This also works for ERPsets if you double-click on the ERP variable.
To see how this works, select the original version of 10_N170 from the Datasets menu and launch EEG Channel operations again. You should still have the 35 equations from before. Near the bottom of the window, click the Save list as button to save the list of equations as a file. Name the file equations.txt . In addition to saving the file, this command puts the filename (including the entire path) in the File: text box. Now, select the check box labeled Send file rather than individual equations , and click Run to execute the routine. Now when you type eegh on the command line, you should see a much shorter line of code that looks something like this:
EEG = pop_eegchanoperator( EEG, '/Users/luck/Ch_10_Scripting/Exercises/equations.txt' , 'ErrorMsg', 'popup', 'KeepChLoc', 'on', 'Warning', 'on' );
Let’s take a closer look at this command. We send the EEG variable as the first parameter. The routine then sends back a new version of this variable as its output. The second parameter is the filename for the equations file (including the whole path, which will be different on your computer). Then we have a sequence of pairs of parameters, which occur in the order parameter name , parameter value . For example, the sequence of parameters 'ErrorMsg', 'popup' tells the routine to set the error message option to a value of popup , which means that error messages will be delivered in popup windows. The next pair of parameters is 'KeepChLoc', 'on', which tells the routine that the option for keeping the channel locations should be on (which you had set from the GUI by checking the box the option labeled Try to preserve channel locations ). The use of pairs of parameters—one for the name of the parameter and one for the value—makes it easier to understand what the parameters are doing. This approach is common in Matlab scripts and functions, and it’s used extensively in EEGLAB and ERPLAB routines.
How do you know what options are available for a command like pop_eegchanoperator ? You can use Matlab’s help feature. There are several ways to access this feature. One common way is to type help pop_eegchanoperator on the command line. Another way is to select the name of the routine in the Command Window (or in a script) and then right-click or control-click on it to get a contextual menu, which contains a Help on Selection item. Try one of these methods to see the help for pop_eegchanoperator . When you do this, Matlab simply shows you all of the comments (lines of text beginning with the % character) at the top of the .m file containing the command (e.g., the pop_eegchanoperator.m file, which contains the code for this routine and is located in a subfolder deep inside the EEGLAB folder that was created when you installed EEGLAB and ERPLAB). This works with any command, including built-in Matlab commands like pwd and EEGLAB and ERPLAB commands like pop_eegchanoperator .
Now it’s time for an embarrassing admission: When I looked up the help for pop_eegchanoperator , I found that it didn’t actually list the optional parameters. In fact, it wasn’t very helpful at all. This is one of the first routines that we wrote for ERPLAB, and I’m guessing that we hadn’t yet established a good workflow that included careful documentation of all the features of each routine. This has now been fixed. And here’s another embarrassing admission: I’ve seen lots of shortcomings in the ERPLAB help information as I’ve worked on this book. I don’t have a good excuse for this. I can tell you that most programmers hate writing documentation—once the code is done, it’s not much fun to write the documentation. But that’s not a good excuse. I’m going to add “go through all the ERPLAB help” to my list of things to do after I finish this book.
But let’s turn lemons into lemonade and use this as an opportunity to learn how to overcome unclear or missing documentation.
Lemonade
There’s an old saying that “When life gives you lemons, make lemonade.” I prefer the snarkier version, “When life gives you lemons, squeeze them into the open wounds of your enemies.”
When I ran into this problem with pop_eegchanoperator , I simply opened the pop_eegchanoperator.m file and looked at the code. I didn’t have to go searching through my file system for this file. I just found pop_eegchanoperator in the Command Window, selected it, control-clicked on it to get a contextual menu, and selected Open Selection . Give that a try to open the pop_eegchanoperator.m file.
If you’re not an expert in Matlab scripting, you may wonder how you’re going to understand the code in the pop_eegchanoperator.m file? You’re interested in finding out about the options, and you know that one of the options is ErrorMsg , so it would make sense to do a search for that string in the file. If you do this, you’ll eventually find this line:
p.addParamValue('ErrorMsg', 'cw', @ischar); % cw = command window
From this line you can infer that a possible value for this parameter is 'cw' , and that this will send the error messages to the Command Window. I f you search some more, you’ll find this set of lines:
if strcmpi(p.Results.ErrorMsg,'popup')
errormsgtype = 1; % open popup window
From these two lines, you can see that the 'popup' value opens a popup window. So, now you’ve learned a strategy for figuring out how a routine works and what the options are (and I’ve learned that I need to put more effort into making sure that the ERPLAB help information is complete).
Now that you’ve learned about the command for EEG Channel Operations, let’s add it to the script we started in the previous exercise. Open Script4.m (if it’s not already open), and save it as MyScript4.m . Copy the pop_eegchanoperator line from eegh history in the Command Window to the clipboard and then paste it into MyScript4.m right after the line that begins with EEG = pop_loadset . Make sure you’ve pasted the version that uses the equations.txt file. Then save the updated MyScript4.m file. (I always save a script before running it—if there’s an error that crashes Matlab, I don’t want to lose the new code that I just added.)
Now quit EEGLAB, type clear all , and run MyScript4.m to see it work. You should see EEGLAB launch and then lots of text appear in the Command Window (mainly from EEG Channel Operations ), including a line that says something like Processing Subject 9 of 9, ID = 10 as each individual subject is being processed. Once the script finishes, you should plot the EEG from one of the datasets, and then you’ll see that you now have bipolar VEOG and HEOG channels.
If you run into an error, make sure that you followed the above instructions carefully, and think about the logic of what you’re doing. You can also consult the Troubleshooting Tips in Appendix 2. The next several sections of this chapter will require you to make various additions to MyScript4.m , and you’ll almost certainly make at least one error at some point. When you do, feel free to make a few unkind comments about my parentage, and then remember that errors are a big part of scripting and that one of the goals of this book is for you to learn how to find and fix problems. Appendix 2 ends with some recommendations for writing code in a way that reduces the likelihood of errors.
After you get the script to run correctly, I want you to spend a moment thinking about how easy it was to add the step for referencing the data to the prior script. Once you had the basic script for looping through the participants and opening their datasets, you just had to run EEG Channel Operations from the GUI, get the history, and copy the command from the history to the appropriate point in the script. Done!
Well, not so fast. You’ll usually want to make a few changes to the command that you copied from the history. For example, the new line of code you added to the script broke my rule about values (such as filenames) being defined as variables at the top of the script. If you moved this script and the equations.txt file to a new folder, the script would break. The next exercise will solve this problem and make some additional improvements.


6.4B: Postnatal Bone Growth
- Differentiate among the zones of development from cartilage to bone in postnatal ossification
Postnatal Ossification
Secondary ossification occurs after birth. It forms the epiphyses of long bones and the extremities of irregular and flat bones.
The diaphysis and both epiphyses of a long bone are separated by a growing zone of cartilage (the epiphyseal plate). When a child reaches skeletal maturity (18 to 25 years of age), all of the cartilage is replaced by bone, fusing the diaphysis and both epiphyses together (epiphyseal closure). This process involves replacing the hyaline cartilage, initially present at the epiphyseal region, with active osteoblasts that deposit bone structural proteins.
Epiphyseal plate : The epiphysis is the rounded end of a long bone located at its joint with adjacent bone(s). Between the epiphysis and diaphysis (the long midsection of the long bone) lies the metaphysis, including the epiphyseal plate (growth plate).
The Role of Osteoblasts
Developing femur : Pictured is part of a longitudinal section of a rabbit’s developing femur, with parts including: a) Flattened cartilage cells; b) Enlarged cartilage cells; c), d) Newly formed bone; e) Osteoblasts; f) Giant cells or osteoclasts; g), h) Shrunken cartilage cells.
Osteoblasts are mononucleate cells that are responsible for bone formation. In essence, osteoblasts are specialized fibroblasts that, in addition to fibroblastic products, express bone sialoprotein and osteocalcin.
Osteoblasts produce a matrix of osteoid that is composed mainly of Type I collagen. Osteoblasts are also responsible for the mineralization of this matrix. Minerals required for mineralization and related processes include zinc, copper, and sodium.
Bone is a dynamic tissue that is constantly being reshaped by osteoblasts and osteoclasts. Osteoblasts produce bone matrix and mineral, and osteoclasts break down the tissue. The number of osteoblasts tends to decrease with age, affecting the balance of the formation and resorption in the bone tissue, and potentially leading to osteoporosis.
During postnatal bone formation, endochondral ossification initiates bone deposition by first generating a structural framework at the ends of long bones, within which the osteoblasts can synthesize a new bone matrix.
Cartilage to Bone
- Zone of reserve cartilage: This region is farthest from the marrow cavity and consists of hyaline cartilage that does not actively transform into bone. Quiescent chondrocytes are found here.
- Zone of cell proliferation: Closer to the marrow cavity, chondrocytes in this region multiply and arrange themselves into longitudinal columns of flattened lacunae.
- Zone of cell hypertrophy: They stop dividing and begin to hypertrophy (enlarge). The walls of the matrix between the lacunae become very thin.
-
Zone of calcification: The region where the cartilagenous matrix begins to calcify.
Minerals are deposited in the matrix between the columns of lacunae, but are not the permanent bone mineral deposits. This acts as a temporary support for the cartilage that would otherwise be weakened due to the breakdown of the lacunae. - Zone of bone deposition (ossification): The walls between the lacunae break down and the chondrocytes die. Each column then becomes a longitudinal channel that is immediately invaded by blood vessels and marrow from the marrow cavity. Osteoblasts begin depositing concentric lamellae of matrix, while osteoclasts dissolve the temporarily calcified cartilage.
The growth in the diameter of bones around the diaphysis occurs through the deposition of bone beneath the periosteum. Osteoclasts in the interior cavity continue to degrade bone until its ultimate thickness is achieved. At this point the rate of formation on the outside and degradation from the inside is constant. This process is termed appositional growth.
Endochondral ossification : The development of the primary and secondary ossification centers.
Key Points
- The epiphyseal plate is the growth zone between the diaphysis (shaft) of the long bone and the epiphysis (end) of the long bone.
- At skeletal maturity, growth ceases when the epiphyses fuse with the diaphyses, indicating that all the cartilage has been replaced with bone and epiphyseal closure has been achieved.
- Osteoblasts are mononucleate cells that are responsible for bone formation.
- Bone is a dynamic tissue that is constantly being reshaped by osteoblasts and osteoclasts.
- Osteoblasts produce a matrix of osteoid, which is composed mainly of Type I collagen.
Key Terms
-
osteocalcin
: Secreted solely by osteoblasts, it is pro-osteoblastic (bone-building),
and is implicated in bone mineralization and calcium ion homeostasis. -
sialoprotein
: A component of mineralized tissues, such as bone, that acts as a nucleus for
the formation of the first apatite crystals. - secondary ossification : A process that occurs after birth, and forms the epiphyses of long bones and the extremities of irregular and flat bones.
- diaphyses : The main or mid section (shaft) of a long bone that is made up of cortical bone.
- epiphyseal closure : The fusion of the epiphysis to the diaphysis.


19.4: Evidentiary Standards You Can Believe
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If [p] is below .02 it is strongly indicated that the [null] hypothesis fails to account for the whole of the facts. We shall not often be astray if we draw a conventional line at .05 and consider that [smaller values of p] indicate a real discrepancy.
– Sir Ronald Fisher (1925)
Consider the quote above by Sir Ronald Fisher, one of the founders of what has become the orthodox approach to statistics. If anyone has ever been entitled to express an opinion about the intended function of p-values, it’s Fisher. In this passage, taken from his classic guide Statistical Methods for Research Workers , he’s pretty clear about what it means to reject a null hypothesis at p<.05. In his opinion, if we take p<.05 to mean there is “a real effect”, then “we shall not often be astray”. This view is hardly unusual: in my experience, most practitioners express views very similar to Fisher’s. In essence, the p<.05 convention is assumed to represent a fairly stringent evidentiary standard.
Well, how true is that? One way to approach this question is to try to convert p-values to Bayes factors, and see how the two compare. It’s not an easy thing to do because a p-value is a fundamentally different kind of calculation to a Bayes factor, and they don’t measure the same thing. However, there have been some attempts to work out the relationship between the two, and it’s somewhat surprising. For example, Johnson (2013) presents a pretty compelling case that (for t-tests at least) the p<.05 threshold corresponds roughly to a Bayes factor of somewhere between 3:1 and 5:1 in favour of the alternative. If that’s right, then Fisher’s claim is a bit of a stretch. Let’s suppose that the null hypothesis is true about half the time (i.e., the prior probability of H 0 is 0.5), and we use those numbers to work out the posterior probability of the null hypothesis given that it has been rejected at p<.05. Using the data from Johnson (2013), we see that if you reject the null at p<.05, you’ll be correct about 80% of the time. I don’t know about you, but in my opinion an evidentiary standard that ensures you’ll be wrong on 20% of your decisions isn’t good enough. The fact remains that, quite contrary to Fisher’s claim, if you reject at p<.05 you shall quite often go astray. It’s not a very stringent evidentiary threshold at all.


11.8: Reading- Geologic Processes and Flowing Water
Streams—any running water from a rivulet to a raging river—complete the hydrologic cycle by returning precipitation that falls on land to the oceans (figure 1). Some of this water moves over the surface and some moves through the ground as groundwater. Flowing water does the work of both erosion and deposition.
Erosion and Deposition by Streams
Erosion by Streams
Flowing streams pick up and transport weathered materials by eroding sediments from their banks. Streams also carry ions and ionic compounds that dissolve easily in the water. Sediments are carried as:
- Dissolved load : Dissolved load is composed of ions in solution. These ions are usually carried in the water all the way to the ocean.
- Suspended load : Sediments carried as solids as the stream flows are suspended load. The size of particles that can be carried is determined by the stream’s velocity (figure 2). Faster streams can carry larger particles. Streams that carry larger particles have greater competence . Streams with a steep gradient (slope) have a faster velocity and greater competence.
- Bed load : Particles that are too large to be carried as suspended load are bumped and pushed along the stream bed as bed load. Bed load sediments do not move continuously. This intermittent movement is called saltation . Streams with high velocities and steep gradients do a great deal of down cutting into the stream bed, which is primarily accomplished by movement of particles that make up the bed load.
- Here is a video of bedload transport.
Stages of Streams
As a stream flows from higher elevations, like in the mountains, towards lower elevations, like the ocean, the work of the stream changes. At a stream’s headwaters , often high in the mountains, gradients are steep (figure 3). The stream moves fast and does lots of work eroding the stream bed.
As a stream moves into lower areas, the gradient is not as steep. Now the stream does more work eroding the edges of its banks. Many streams develop curves in their channels called meanders (figure 4).
As the river moves onto flatter ground, the stream erodes the outer edges of its banks to carve a floodplain , which is a flat level area surrounding the stream channel (figure 5).
Base level is where a stream meets a large body of standing water, usually the ocean, but sometimes a lake or pond. Streams work to down cut in their stream beds until they reach base level. The higher the elevation, the farther the stream is from where it will reach base level and the more cutting it has to do.
Stream Deposition
As a stream gets closer to base level, its gradient lowers and it deposits more material than it erodes. On flatter ground, streams deposit material on the inside of meanders. Placer mineral deposits, described in the Earth’s Minerals chapter, are often deposited there. A stream’s floodplain is much broader and shallower than the stream’s channel. When a stream flows onto its floodplain, its velocity slows and it deposits much of its load. These sediments are rich in nutrients and make excellent farmland (figure 6).
A stream at flood stage carries lots of sediments. When its gradient decreases, the stream overflows its banks and broadens its channel. The decrease in gradient causes the stream to deposit its sediments, the largest first. These large sediments build a higher area around the edges of the stream channel, creating natural levees (figure 7).
When a river enters standing water, its velocity slows to a stop. The stream moves back and forth across the region and drops its sediments in a wide triangular-shaped deposit called a delta (figure 8).
If a stream falls down a steep slope onto a broad flat valley, an alluvial fan develops (figure 9). Alluvial fans generally form in arid regions.
Contributors and Attributions
- 10.1: Water Erosion and Deposition. Provided by : CK-12. Located at : http://www.ck12.org/book/CK-12-Earth-Science-For-High-School/section/10.1/ . License : CC BY-NC: Attribution-NonCommercial


4.1: Stoichiometry
What is Stoichiometry
Stoichiometry (stoi·chi·om·e·try /ˌstɔɪkiˈɒmɪtri/) is the study of the quantities of substances and energy consumed and produced in chemical reactions.
The basis of the stoichiometric calculations is the law of conservation of mass which states that the mass is neither created nor destroyed in a chemical reaction. Another form of the law states that atoms are neither created nor destroyed in a chemical reaction. It is the basis of stoichiometric calculations that are described in this chapter.
It can be concluded from the law of mass action that atoms of each element and their masses are the same in reactants and products. A balanced chemical equation shows atoms of each element and the total mass of reactants equal to that in the product, as illustrated in Fig. 4.1.1. The number of atoms and molecules is related to their quantity in moles through Avogadro's number. The mole, in turn, is related to the mass of the substance through molar mass in grams. These relationships are described in the next sections.
Atomic mass
Atomic mass is the weighted average of the masses of the isotopes present in a natural sample of the element, as explained with an example calculation in section 2.3. The mass of a single atom or molecule is expressed in the atomic mass unit (amu), which is equal to \(\frac{1}{2}\)th of the mass of \(\ce{^12_6C}\) isotope of carbon that is unbound.
Atomic mass is listed in a periodic table as a number below the symbol and name of the element, as illustrated in Fig. 4.1.2. Atomic mass is listed as a number without a unit because it is the mass of an atom in amu and it is also the mass of one mole (molar mass) of the atom in grams. Molar mass is often used in stoichiometry calculation as explained in the next sections.


13.1.2: Dynamic Theory of Tides
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The Equilibrium Theory of tides predicts that each day there will be two high and two low tides, each one occurring at the same time day after day, with each pair producing tides of similar heights. While this view provides a basic explanation for the primary forces that generate the tides, it does not take into account such variables as the effects of the continents, the depth of the water, and many other factors. In all, there are almost 400 variables that must be incorporated into predicting the tides! The Dynamic Theory of tides takes these other factors into account, and shows that the tides are much more complicated and variable from place to place than the Equilibrium Theory would suggest. For example, some areas receive only one high and one low tide per day (see section 11.3). Furthermore, the tidal range varies greatly across the globe; in the Mediterranean Sea, there can be a difference of only 10 cm between high and low tides, while the Bay of Fundy in Canada experiences a tidal range of up to 17m (56 ft) every day (Figure \(\PageIndex{1}\)).
Examination of any tide chart will show that the tides don’t occur at same time each day; in fact, each tidal peak occurs about 50 minutes later than it did in the previous day. This is due to the orbit of the moon around the Earth. Imagine a high tide that occurs at a particular location (X) at 1:00 pm (Figure \(\PageIndex{2}\)). The high tide occurs as location X moves through the bulge of water facing the moon. It will take the Earth 24 hours to complete one revolution, to bring location X back to site of the water bulge that caused that high tide. However, during those 24 hours, the moon has also moved as it orbits the Earth, so the high tide bulge has moved beyond its original location. The Earth thus has to rotate an additional distance for location X to reach the bulge and experience that same high tide. Because it takes the moon about 28 days to orbit the Earth, the moon gets “ahead” of the Earth’s rotation by about 50 minutes per day. Therefore, it takes location X 24 hours and 50 minutes to rotate through the same tidal bulge, and as a result, the tidal peaks occur about 50 minutes later each day. In our example, an afternoon high tide at 1:00 pm on one day would be followed by a high tide at about 1:50 pm the following day. This 24 hour and 50 minute cycle is referred to as a tidal day .
The motion of the moon impacts the tidal cycles in other ways. As the moon orbits the Earth, its orbital plane is at an angle relative to the rotational plane of Earth. This angle, or declination , means than the moon fluctuates between an angle of 28.5 o north of the equator, to 28.5 o south of the equator roughly every two weeks (the cycle from maximum to minimum and back takes about 27 days). Figure \(\PageIndex{3}\) illustrates a case where the moon is at its maximum declination 28.5 o north of the equator, creating its corresponding tidal maxima. A point on the Earth at the latitude indicated by the red line would experience two high tides as it rotated through 24 hours, at points A and B. But the two high tides would not be of equal heights; the high tide at A would be higher than the high tide at B. This helps create a mixed semi-diurnal tide; two high tides of different heights per day (see section 11.3).
Finally, the continents and the bottom topography of the oceans have an impact on the tides that are experienced in an area. Because the tides are essentially waves with extremely long wavelengths extending halfway across the Earth, they behave as shallow water waves, and they are influenced and refracted by the bottom contours, leading to regional tidal variations. When the tidal crests encounter land, they are are reflected, and the wave moves back out to sea, theoretically until it encounters another continent on the opposite side of the ocean basin. The crest is once again reflected, and the water oscillates back and forth as a standing wave across the ocean basin. However, because of the scale over which these tidal waves move, we must take into account the influence of the Coriolis Effect. As the tidal crest is reflected back across the ocean basin, its path is deflected by the Coriolis force; to the right in the Northern Hemisphere, and to the left in the Southern Hemisphere (see section 9.1). Using the Northern Hemisphere as an example, imagine a tidal crest that has reached land on the western side of an ocean basin. It would have a tendency to be reflected and move across the basin towards the east. But the Coriolis force deflects the movement to the right, causing the crest to instead head south. When the crest hits land in the south, it would now tend to reflect towards the north, but once again the Coriolis deflection to the right kicks in, and the wave instead moves to the east. From the east the reflected wave is deflected to the north, and so on. The result of all of this is that instead of a simple standing wave moving back and forth across the ocean, the tidal crest follows a circular pattern around the ocean basin, counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. This is analogous to shaking a pan full of water in a circular manner, and watching the water follow a similar circular path as it sloshes around inside. This large scale circular rotation pattern of tides is called amphidromic circulation (Figure \(\PageIndex{4}\)). The rotation occurs around a central amphidromic point or node , that shows little tidal variation, while the largest tidal ranges occur on the edges of the circulation pattern. In Figure \(\PageIndex{4}\) the amphidromic points are indicated by the dark blue areas where the white lines converge, like spokes from a bicycle wheel, and the dark red and brown areas show the regions of maximum tidal heights. The tidal maxima will rotate around the amphidromic points, taking about 12 hours for a complete rotation, leading to two high and two low tides per day in many places. If a tidal maximum is occurring along one of the white lines in Figure \(\PageIndex{4}\) at a certain time in the Northern Hemisphere, one hour later that high tide will have moved to the white line to the left (counterclockwise), and so on until it completes a rotation. In the Southern Hemisphere, the tide will move to the line to the right for clockwise rotation.
The result of all of these variables is that the tides will not always occur twice each day, at the same time and with equal heights as the Equilibrium Theory of tides may suggest. Instead, each region of the oceans has a unique set of factors that contribute to the types of tides it will experience. The major types of tides are discussed in the next section.


7.3G: Vitamin B₉ (Folic acid)
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Folic acid, another form of which is known as folate, is one of the B vitamins. The recommended daily intake level of folate is 400 micrograms from foods or dietary supplements. Folic acid is used to treat anemia caused by folic acid deficiency. It is also used as a supplement by women during pregnancy to prevent neural tube defects (NTDs) in the baby. Low levels in early pregnancy are believed to be the cause of more than half of babies born with neural tube defects. More than 50 countries use fortification of certain foods with folic acid as a measure to decrease the rate of NTDs in the population. Long term supplementation is also associated with small reductions in the risk of stroke and cardiovascular disease. It may be taken by mouth or by injection.
Skeletal formula
There are no common side effects. It is not known whether high doses over a long period of time are of concern. There are concerns that large amounts of folic acid might hide vitamin B12 deficiency. It is essential for the body to make DNA, RNA, and metabolise amino acids which are required for cell division.[8] As humans cannot make folic acid, it is required from the diet, making it an essential vitamin.
Not consuming enough folate can lead to folate deficiency. This may result in a type of anemia in which low numbers of large red blood cells occur. Symptoms may include feeling tired, heart palpitations, shortness of breath, open sores on the tongue, and changes in the color of the skin or hair. Deficiency in children may develop within a month of poor dietary intake.[13] In adults normal total body folate is between 10,000–30,000 micrograms (µg) with blood levels of greater than 7 nmol/L (3 ng/mL).[8]
Folic acid was discovered between 1931 and 1943.[14] It is on the World Health Organization's List of Essential Medicines, the most effective and safe medicines needed in a health system.[15] The wholesale cost of supplements in the developing world is between 0.001 and 0.005 USD per dose as of 2014.[16]The term "folic" is from the Latin word folium, which means leaf. Folates occur naturally in many foods especially dark green leafy vegetables and liver.[8]
Pregnancy and Fertility
Folic acid intake during pregnancy has been linked to a lessened risk of neural tube defects. Likewise, a meta-analysis of folic acid supplementation during pregnancy reported a 28% lower risk of newborn congenital heart defects. The United States Preventive Services Task Force recommends folic acid supplementation for all women able to become pregnant.
Devakamar reviewed long-term outcomes for live births that did not involve neural tube defects and fortification with folic acid did not have an impact on childhood survival, growth, body composition, or cognitive outcomes. Prenatal supplementation did not appear to reduce the risk of pre-term births. And there does not appear to be a correlation between maternal folic acid supplementation and an increased risk for asthma in the child.
Folate is necessary for fertility in both men and women. It contributes to spermatogenesis. Therefore, it is necessary to receive sufficient amounts through the diet to avoid subfertility. Also, polymorphisms in genes of enzymes involved in folate metabolism could be one reason for fertility complications in some women with unexplained infertility.
Heart Disease, Stroke and Cancer
Taking folic acid over years reduced the risk of cardiovascular disease by 4%, where another study found it did not affect cardiovascular disease, even while reducing homocysteine levels.
Long-term supplementation with folic acid reduced the risk of stroke by 10%, which may be due to the role folate plays in regulating homocysteine concentration.The reviews indicate the risk of stroke appears to be reduced only in some individuals, but a definite recommendation regarding supplementation beyond the current RDA has not been established for stroke prevention. Asian populations had greater protection against stroke with folate supplementation than did European or North American subjects.
Observed stroke reduction is consistent with the reduction in pulse pressure produced by folate supplementation of 5 mg per day, since hypertension is a key risk factor for stroke. Folic supplements are inexpensive and relatively safe to use, which is why stroke or hyperhomocysteinemia patients are encouraged to consume daily B vitamins including folic acid.
Studies on folic acid intake from food and folate supplementation with regards to cancer risk are based on the adequacy of chronic intake. Chronically insufficient intake of folic acid (below the recommended level of 400 micrograms per day) may increase the risk of colorectal, breast, ovarian, pancreas, brain, lung, cervical, and prostate cancers. Other studies showed that excessive dietary supplementation with synthetic folate may increase the risk of certain cancers, in particular prostate. A 2017 review found no relationship between taking folate supplements and cancer risk.
Dietary Reference Intake
Because of the difference in bioavailability between supplemented folic acid and the different forms of folate found in food, the dietary folate equivalent (DFE) system was established. One DFE is defined as 1 μg of dietary folate, or 0.6 μg of folic acid supplement.
| Age | Infants (AI) | Infants (UL) | Adults (RDA) | Adults (UL) | Pregnant women (RDA) | Pregnant women (UL) | Lactating women (RDA) | Lactating women (UL) |
|---|---|---|---|---|---|---|---|---|
| 0–6 months | 65 | None set | – | – | – | – | – | – |
| 7–12 months | 80 | None set | – | – | – | – | – | – |
| 1–3 years | – | – | 150 | 300 | – | – | – | – |
| 4–8 years | – | – | 200 | 400 | – | – | – | – |
| 9–13 years | – | – | 300 | 600 | – | – | – | – |
| 14–18 | – | – | 400 | 800 | 600 | 800 | 500 | 800 |
| 19+ | – | – | 400 | 1000 | 600 | 1000 | 500 | 1000 |
For U.S. food and dietary supplement labeling purposes the amount in a serving is expressed as a percent of Daily Value (%DV). For folic acid labeling purposes 100% of the Daily Value was 400 μg. As of the May 2016 update it was kept unchanged at 400 μg. A table of the pre-change adult Daily Values is provided at Reference Daily Intake. Food and supplement companies have until 28 July 2018 to comply with the change.
Dietary Sources
Folate naturally occurs in a wide variety of foods, including vegetables (particularly dark green leafy vegetables), fruits and fruit juices, nuts, beans, peas, dairy products, poultry and meat, eggs, seafood, grains, and some beers. Avocado, beetroot, spinach, liver, yeast, asparagus, and Brussels sprouts are among the foods with the highest levels of folate. Folic acid is added to grain products in many countries, and in these countries, fortified products make up a significant source of the population's folic acid intake. Because of the difference in bioavailability between supplemented folic acid and the different forms of folate found in food, the dietary folate equivalent (DFE) system was established. 1 DFE is defined as 1 μg of dietary folate, or 0.6 μg of folic acid supplement. This is reduced to 0.5 μg of folic acid if the supplement is taken on an empty stomach. bFolate naturally found in food is susceptible to high heat and ultraviolet light, and is soluble in water. It is heat-labile in acidic environments and may also be subject to oxidation. Some meal replacement products do not meet the folate requirements as specified by the RDAs.
Food Fortification
Folic acid fortification is a process where folic acid is added to flour with the intention of promoting public health through increasing blood folate levels in the populace. In the USA, food is fortified with folic acid, only one of the many naturally occurring forms of folate, and a substance contributing only a minor amount to the folates in natural foods. Since the discovery of the link between insufficient folic acid and neural tube defects, governments and health organizations worldwide have made recommendations concerning folic acid supplementation for women intending to become pregnant.
Fortification is controversial, with issues having been raised concerning individual liberty, as well as the health concerns described in the Toxicity section above. In the USA, there is concern that the federal government mandates fortification, but does not provide monitoring of potential undesirable effects of fortification. 76 countries worldwide (inclduing the USA) require mandatory folic acid fortification of at least one major cereal grain, with nearly all fortifying at least wheat flour, according to November 2013 data from the Flour Fortification Initiative.
Folate Deficiency
Folate deficiency can be caused by unhealthy diets that do not include enough fruits and vegetables, diseases in which folic acid is not well absorbed in the digestive system (such as Crohn's disease or celiac disease), some genetic disorders that affect levels of folate, and certain medicines (such as phenytoin, sulfasalazine, or trimethoprim-sulfamethoxazole). Folate deficiency is accelerated by alcohol consumption.
Folate deficiency may lead to glossitis, diarrhea, depression, confusion, anemia, and fetal neural tube defects and brain defects (during pregnancy). Other symptoms include fatigue, gray hair, mouth sores, poor growth, and swollen tongue. Folate deficiency is diagnosed by analyzing CBC and plasma vitamin B 12 and folate levels. CBC may indicate megaloblastic anemia but this could also be a sign of vitamin B 12 deficiency. A serum folate of 3 μg/L or lower indicates deficiency. Serum folate level reflects folate status but erythrocyte folate level better reflects tissue stores after intake. Serum folate reacts more rapidly to folate intake than erythrocyte folate. An erythrocyte folate level of 140 μg/L or lower indicates inadequate folate status. Increased homocysteine level suggests tissue folate deficiency but homocysteine is also affected by vitamin B 12 and vitamin B 6 , renal function, and genetics.
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