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Chapter 10: Problem 11

Polymorphonuclear (PMN) phagocytes (white blood cells) are generally the first defense mechanism employed in the body in response to bacterial inva- (a) Write a set of equations to describe the motions and interactions of microbes \(b\) and phagocytes \(c\). (b) Additional assumptions made were that \(\begin{aligned} f(b)=& \frac{k_{z} b}{1+b / K_{i}}, \quad d(b, c)=\frac{-k_{d} b c}{K_{b}+b}, \\ & s A(c, b)=h_{0} \frac{A}{V} C_{b}\left(1+\frac{h_{1}}{h_{0}} b\right) \\ \text { where } k_{z}=& \text { bacterial growth rate constant, } \\ k_{a}=\text { phagocytic killing rate constant, } \\ h_{0}=\text { rate of emigration from venules when inflammation is ab- } \\ & \text { sent, } \\ h_{1} &=\text { inflammation- enhanced emigration rate, } \\ / V &=\text { ratio of venule wall-surface area to tissue volume, } \\ &=\text { phagocyte density in the venules. } \end{aligned}\) \(k_{d}=\) phagocytic killing rate constant, \(h_{0}=\) rate of emigration from venules when inflammation is absent, \(h_{1}=\) inflammation-enhanced emigration rate, \(A / V=\) ratio of venule wall-surface area to tissue volume, \(C_{b}=\) phagocyte density in the venules. (1) Explain the meaning of these assumptions. (2) Define \(K_{i}\) and \(K_{b}\) and give dimensions of all parameters above. (c) By dimensional analysis, it is possible to reduce the equations to the following form: $$ \begin{aligned} &\frac{\partial v}{\partial \tau}=\rho \frac{\partial^{2} v}{\partial \xi^{2}}+\frac{\gamma v}{1+v}-\frac{u v}{\kappa+v} \\ &\frac{\partial u}{\partial \tau}=\frac{\partial^{2} u}{\partial \xi^{2}}-\delta \frac{\partial}{\partial \xi}\left(u \frac{\partial v}{\partial \xi}\right)+\alpha(1+\sigma v-u) . \end{aligned} $$ (d) Show that the equations in part (c) have two types of uniform steady-state solutions: (1) \(v=0, \quad u=1\). (2) \(v>0, u=1+\sigma v\). Identify the biological meaning of these steady states.

Short Answer

Expert verified
1) Write equations for microbes' motion and interactions. 2) Define the given functions and parameters. 3) Reduce to simplified form. 4) Steady states are (0, 1) and (v > 0, 1 + σv).

Step by step solution

01

- Write the Equations Describing Motions and Interactions

Let's denote the population of microbes by \(b\) and phagocytes by \(c\). One common model for their interactions including growth and killing is given by the equations:\[ \frac{db}{dt} = f(b) - d(b, c) \]\[ \frac{dc}{dt} = sA(c, b) - k_{a}c \].
02

- Specify the Functions Based on Assumptions

From the problem, the specific forms of the functions are:\[ f(b) = \frac{k_{z} b}{1 + b/K_{i}} \]\[ d(b, c) = \frac{k_{d} bc}{K_{b} + b} \]\[ sA(c, b) = h_{0} \frac{A}{V} C_{b} \left(1 + \frac{h_{1}}{h_{0}} b\right) \]These specify bacterial growth, bacterial killing rate by phagocytes, and the rate of phagocyte population change respectively.
03

- Explain the Assumptions

(1) The assumptions can be interpreted as follows:- \(f(b)\) models bacterial growth which saturates at higher densities.- \(d(b, c)\) represents the bacterial killing rate by phagocytes, which increases with more bacteria and phagocytes but also saturates.- \(sA(c, b)\) models the rate of phagocyte recruitment, which increases both due to normal recruitment (\(h_{0}\) term) and inflammation (\(h_{1}\) term).
04

- Define Parameters and Dimensions

(2) The parameters \(K_{i}\) and \(K_{b}\) represent the bacterial population constant for growth saturation and the bacterial population constant for phagocytic killing saturation, respectively. The dimensions of the parameters are as follows:\(k_{z} : [T^{-1}]\)\(k_{d} : [T^{-1}]\)\(h_{0} : [T^{-1}]\)\(h_{1} : [T^{-1}]\)\(A/V : [L^{-1}]\)\(C_{b} : [L^{-3}]\)\(K_{i} : [N] (population)\)\(K_{b} : [N] (population)\)
05

- Dimensional Reduction

(c) By non-dimensionalizing the equations using suitable scaling for time, space, and population densities, the equations can be reduced to:\[ \frac{\partial v}{\partial \tau} = \rho \frac{\partial^{2} v}{\partial \xi^{2}} + \frac{\gamma v}{1+v} - \frac{u v}{\kappa+v} \]\[ \frac{\partial u}{\partial \tau} = \frac{\partial^{2} u}{\partial \xi^{2}} - \delta \frac{\partial}{\partial \xi} \left( u \frac{\partial v}{\partial \xi} \right) + \alpha(1 + \sigma v - u) \].
06

- Solve for Steady States

(d) A uniform steady-state solution occurs where \( \frac{\partial v}{\partial \tau} = 0 \) and \( \frac{\partial u}{\partial \tau} = 0 \):For the given equations, setting time derivatives to zero and solving for \(v\) and \(u\) yields:1) \(v = 0, u = 1\): This corresponds to the absence of microbes and a normal phagocyte level.2) \(v > 0, u = 1 + \sigma v\): This implies a positive and stable microbe population with a corresponding higher phagocyte level due to inflammation response.
07

- Interpret Biological Steady States

The first steady state (\(v = 0, u = 1\)) indicates a healthy state with no infection. The second steady state (\(v > 0, u = 1 + \sigma v\)) indicates an ongoing infection with a balanced increase in phagocyte response to keep the infection in check.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Phagocytes
Phagocytes are a crucial part of the immune system. They are specialized cells that engulf and digest microbes and foreign particles. Among the first responders to infection, phagocytes identify, attack, and neutralize pathogens by consuming them—a process called phagocytosis. This response is particularly vital in the fight against bacterial infections. The effectiveness of phagocytes can be influenced by factors such as the presence of inflammation and the density of phagocytes in the blood vessels. Emigration from capillaries into affected tissues, especially during inflammation, enhances their availability and effectiveness.
Bacterial Growth
Bacteria can multiply under favorable conditions, leading to exponential population growth. The growth rate depends on several factors, such as nutrient availability and environmental conditions. In mathematical models, this growth is often represented with functions that show saturation at higher bacterial densities. For instance, the bacterial growth rate may follow a logistic model where the rate decreases as the population approaches a carrying capacity, due to limitations like nutrient depletion. This saturation can be modeled as:
  • The growth rate function: \( f(b) = \frac{k_{z} b}{1 + b/K_{i}} \) where \( b \) is the bacterial density.
  • Here, \( k_{z} \) represents the bacterial growth rate constant, and \( K_{i} \) is a constant indicating the population level at which the growth rate is half its maximum value.
This approach ensures the growth rate diminishes as the bacteria population becomes too large to be sustained by the environment.
Dimensional Analysis
Dimensional analysis is a method used in mathematical modeling to simplify complex equations. It involves rescaling variables to dimensionless forms, revealing the underlying structure of the problem. This process often reduces the number of parameters, making the model easier to analyze. In the context of bacterial-phagocyte interactions, dimensional analysis can help reflect the balance between growth, killing mechanisms, and recruitment dynamics.
  • For example, changing variables from real-world units (like time and population size) to dimensionless forms (like \( \tau \) for time and \( v \) for bacterial density) provides a clearer view of the system's behavior across scales.
  • The primary goal is to derive simpler, yet accurate, equations that capture the essential dynamics of the system.
This approach can elucidate fundamental relationships, such as the ratio of bacterial growth to phagocyte killing efficiency.
Steady-State Solutions
Steady-state solutions refer to conditions where system variables no longer change over time, indicating a balance within the model. In the case of bacterial and phagocyte dynamics, steady states provide insight into possible outcomes of infection and immune response. Finding steady-state solutions involves setting the time derivatives to zero and solving the resulting algebraic equations.
  • Two key steady-state solutions are identified: \( v = 0, u = 1 \) and \( v > 0, u = 1 + \sigma v \).
  • The first solution represents an infection-free state with normal phagocyte levels, indicating no bacteria present and a healthy immune system.
  • The second solution depicts a controlled infection where bacteria persist but are managed by an elevated phagocyte response, reflecting an active immune response to keep the infection in check.
Understanding these states helps in predicting the possible long-term behavior of bacteria and phagocyte populations under different conditions.

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Most popular questions from this chapter

Plant-herbivore systems and the qualify of the vegetarion. In problem 17 of Chapter 3 and problem 20 of Chapter 5 we discussed models of plant-herbivore interactions that considered the quality of the vegetation. We now further develop a mathematical framework for dealing with the problem. We shall assume that the vegetation is spatially uniform but that there is a variety in the quality of the plants. By this we mean that some chemical or physical plant trait \(q\) governs the success of herbivores feeding on the vegetation. For example, \(q\) might reflect the succulence, nutritional content, or digestibility of the vegetation, or it may signify the degree of induced chemical substances, which some plants produce in response to herbivory. We shall be primarily interested in the mutual responses of the vegetation and the herbivores to one another. (a) Define \(q(t)=\) quality of the plant at time \(t\), \(h(t)=\) average number of herbivores per plant at time \(t\). Reason that equations describing herbivores interacting with a (single) plant might take the form $$ \frac{d q}{d t}=f(q, h), \quad \frac{d h}{d t}=g(q, h)=h r(q, h) \text {. } $$ What assumptions underly these equations? (b) We wish to define a variable to describe the distribution of plant quality in the vegetation. Consider \(p(q, t)=\) biomass of the vegetation whose quality is \(q\) at time \(t\). Give a more accurate definition by interpreting the following integral: $$ \int_{q}^{p+d p} p(q, t) d q=? $$ (c) Show that the total amount of vegetation and total quality of the plants at time \(t\) is given by $$ \begin{aligned} &P(t)=\int_{0}^{\infty} p(q, t) d q, \\ &Q(t)=\int_{0}^{\infty} q p(q, t) d q . \end{aligned} $$ What would be the average quality \(\bar{Q}(t)\) of the vegetation at time \(t\) ? (d) Suppose that there is no removal (death) or addition of plant material. Write down an equation of conservation for \(p(q, t)\) that describes how the distribution of quality changes as herbivory occurs (Hint: Use an analogy similar to that of Sections \(10.8\) and 10.9.) (e) Suppose that the herbivores are only affected by the average plant quality, \(Q(t)\). What would this mean biologically? What would it imply about the equation for \(d h / d t\) ? (f) Further suppose that the function \(f(q, h)\) is linear in \(q\). (Note: This is probably an unrealistic assumption, but it will be used to illustrate a point.) Assume that $$ f(q, h)=f_{1}(h)+q f_{2}(h) . $$ Interpret the meanings of \(f_{1}\) and \(f_{2}\). *(g) Show that the model thus far can be used to conclude that an equation for the average quality of the vegetation is $$ \frac{d \bar{Q}}{d t}=f_{1}(h)+f_{2}(h) \bar{Q} . $$ [Use the assumptions in parts (d) and (f), the equation you derived in (d), and integration by parts.] (h) Explore what this model would imply about average quality and average number of herbivores per plant if \(f\) and \(r\) are given by $$ \begin{gathered} f(q, h)=K_{1}-K_{2} q h\left(h-h_{0}\right) \\ r(q, h)=K_{3}\left(1-K_{4} h / \bar{Q}\right) \end{gathered} $$

Suggest a (set of) partial differential equations to describe the following processes: (a) A predator-prey system in which both species move randomly in a one. dimensional setting. (b) A predator-prey system in which the predator moves towards higher prey densities and the prey moves towards lower predator densities. (c) A pair of reacting and diffusing chemicals such that species 1 activates the formation of both substances and species 2 inhibits the formation of both substances. Design your model so that it has a homogeneous steady state that is stable. (d) A population of cells that secretes a chemical substance. Assume that the cells orient and move chemotactically in gradients of this chemical and also more randomly. Further assume that the chemical diffuses and is gradually broken down at some rate.

Many of Skellam's arguments are based directly on random-walk calculations, not derived from the continuous PDEs such as equation (1). In his original article he defines the following: \(a^{2}=\) mean square dispersion per generation, \(R=\) radial distance from point at which population was released, \(\lambda=\) growth rate of the population, \(n=\) number of elapsed generations, \(p=\) proportion of the population lying outside a circle of radius \(R\) after \(n\) generations. He proves that $$ p=\exp \frac{-R^{2}}{n a^{2}} $$ (a) If the population growth is described by $$ N_{n+1}=\lambda N_{n} $$ and initially there is just a single individual, show that the population in the \(n\)th generation is \(N_{n}=\lambda^{n}\). (b) Now consider a region that contains all but a single individual. Show that the radius of this region at the \(n\)th generation is given by $$ R_{s}=\left(n a^{2} \ln N_{*}\right)^{1 / 2} $$ [Hint: Why is it true that \(1 / N_{n}=\exp \left(-R^{2} / n a^{2}\right)\) holds for this radius?] (c) Use parts (a) and (b) to show that $$ R_{n}=n a(\ln \lambda)^{1 / 2} $$ He proves that $$ p=\exp \frac{-R^{2}}{n a^{2}} $$ (a) If the population growth is described by $$ N_{n+1}=\lambda N_{s} $$ and initially there is just a single individual, show that the population in the \(n\)th generation is \(N_{n}=\lambda^{n}\). (b) Now consider a region that contains all but a single individual. Show that the radius of this region at the \(n\)th generation is given by $$ R_{s}=\left(n a^{2} \ln N_{0}\right)^{1 / 2} $$ [Hint: Why is it true that \(1 / N_{n}=\exp \left(-R^{2} / n a^{2}\right)\) holds for this radius?] (c) Use parts (a) and (b) to show that $$ R_{n}=n a(\ln \lambda)^{1 / 2} $$ (d) Show that, save for a proportionality factor, this result agrees with the rate of spread of a population given by equation (4).

Segel et al. (1977) calculated the motility coefficients based on equation (6) (where \(r=0\) ) as follows. A capillary tube (cross-sectional area \(=A\) ) is filled with fluid. At time \(t=0\) the open end is placed in a bacterial suspension of concentration \(C_{0}\) and removed at time \(t=T\). The number of bacteria in the tube is counted. Motility \(\mu\) is then computed as follows: $$ \mu=\frac{\pi N^{2}}{4 C \hat{B} A^{2} T} $$ (a) Give justification for or derive their formula. (b) Suppose that (1) the radius of the capillary is approximately \(0.01 \mathrm{~cm}\), (2) the bacterial suspension contains a density \(C_{0}=\left(1 / 7 \times 10^{-7}\right)\) bacteria per milliliter, and (3) the following observations (from Segel et al., 1977 ) are made: $$ \begin{aligned} &\begin{array}{cc} T \text { (min) } & N \text { (bacteria) } \\ \hline 2 & 1800 \\ 5 & 3700 \\ 10 & 4800 \\ 12.5 & 5500 \\ 15 & 6700 \\ \hline \end{array}\\\ &\text { What do you conclude about } \mu ? \end{aligned} $$

Other modeling problems related to tissue cultures (a) Suppose a tissue culture is grown in a chemostat, with constant outflow at some rate \(F\). Give a set of equations to describe the problem. (b) If particles of size \(r_{\max }\) always break apart into \(n\) identical particles of size \(r_{n}\), how would the problem be formulated? *(c) Suppose now that pellets can diminish in size due to shaving off of minute pieces (such as single cells) as a result of friction or turbulence. Assume this takes place at a rate proportional to the pellet radius. How would you model this effect in the following two situations? (1) All minute pieces can then grow into bigger pieces (participate in the overall growth). (2) All fragments die.
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