
MBW:Complement Modeling
From MathBio
An investigation into the results of "The Alternative Pathway of Complement" Michael K. Pangburn & Micheal J. MullerEberhard by Matanya Horowitz and Colin Peterson. A review of the article was written in an earlier project.
Summary
 Research on the part of the biomedical community has led to the creation of a model illustrating the Alternative Pathway of Complement. (See APPM4390: Immune Complement for review of this research). This model qualitatively predicts a set of behaviours that correspond to observed experimental data. However, this model has not been quantitatively compared against the available data, and so has not been strongly validated. This investigation looks to remedy this by constructing the model is Matlab's Simbiology, using parameters gleaned from "The Alternative Pathway of Complement"^{[1]}, and to perform a series of stochastic simulations (see stochastic epidemic modeling section) to verify the model.
 To expand upon the results in the paper, we have investigated a variety of possible configurations in the model to find what parameter set yields results most closely matching those gained experimentally.
 Upon examining the results of the paper, it became apparent that the C3b formation distribution's shape over time closely resembled that of a Poisson process. We have therefore examined the ability to model the reaction as a Poisson process, and its possible ramifications.
Background
 The human body has a large battery of defense mechanisms used to protect itself against foreign infectious agents. Among these is the complement system, a biochemical cascade of reactions that is part of the body's innate defense system. The complement system is divided up into two pathways, the classical pathway and the alternative pathway. It is the alternative pathway that is the focus of a paper, The Alternative Pathway of Complement by Michael K. Pangburn and Hans J. MullerEberhard, which contains the results that we wish to emulate.
 In the body, the liver produces the base protein of the complement system, C3. These C3 proteins can break apart spontaneously in water into two smaller proteins, C3b and C3a. This C3b, as well as another protein, Factor B, can then serve as catalysts for this reaction. If a pathogen is present in the body, C3b can bind to the membrane of the pathogen. If this occurs the C3b, while attached to the pathogen membrane, can bind with Factor B, to form the protein C3bB. This new complex is then cleaved into two proteins designated Ba and Bb. Bb remains attached to C3b on the surface of the pathogen, to form C3bBb, also called C3 Convertase, while Ba floats off.
Figure 1: Complement alternative pathway overview. Generated using SimBiology Matlab package.
 The C3 Convertase, now attached to a pathogen, serves as a catalyst and rapidly breaks apart free floating C3 proteins into C3a and C3b. This renews the detection cycle and results in a positive feedback loop that deposits large quantities of C3bBb on the surface of the pathogen.
 The presence of C3 Convertase and C3b also allow for a larger structure, C3bBbC3b to form. This protein can split apart another protein of the complement system, C5, into C5a and C5b. These proteins initiate a cascade of reactions that result in C6, C7, C8, and C9 proteins forming a large structure together, called the membrane attack complex (MAC). The MAC complex forms on the surface of a pathogen and once it is large enough, it 'punches a whole' into the pathogen. This allows for lysis of the pathogen as the the whole thats created by the MAC allows for free diffusion to occur from between the inside and the outside of the pathogen.
Biological Phenomenon

Figure 2: Distribution of fluorescence ( C3b) on pathogens over time ^{[1]} As mentioned previously, the formation of C3bBb on the surface of the pathogen initializes a positive feedback loop. This feedback loop has important ramifications for the body since it is the primary mechanism by which the complement gets a sufficient concentration to begin the formation of the MAC complex. It is this process of formation that is examined in the aforementioned paper, which contains experimental results, gained through the use of a fluorescence activated cell sorter, showing the formation of C3bBb on the pathogen surface over time (as shown in Figure 2).
 As may be seen in the experimental results, the concentration of the C3bBb, initially starting at zero, begins only slowly gaining C3bBb. However, at around 1.4 minutes, we see a large shift to the right as the positive feedback loop begins to have a larger effect. At later points in time, the concentrations resemble a Gaussian curve, appearing symmetrical as the C3bBb reaches some maximum coverage threshold of the pathogen.
 What is significant about the results is the relatively short time between the small distribution of C3bBb at times less than 1.2 minutes to the rapid saturation of the pathogen, occuring after 1.6 minutes. This is indicative of a highly effective positive feedback process which matches nicely with intuition. The body, once identifying a pathogen, would like to destroy it as quickly as possible. Therefore, it seeks to cover the pathogen in as small a time as possible, a behavior that the positive feedback loop provides.
 It is this positive feedback phenomenon that is the subject of this investigation. If it is indeed the feedback loop that allows for the C3bBb to be so rapidly deposited than the parameters associated with the reaction as well as the chemicals that it depends upon will be important aspect of the alternative pathway.
Computational Mathematical Model

Figure 3: Simbiology Model Using Matlab's Simbiology, it is possible to create a model of the alternative pathway in order that the concentration of the various proteins of the alternative pathway can be viewed over time. The specific model that was used is shown in Figure 3. Similar to that shown in Figure 1, this model is truncated since it is the reaction up to C3bBb that is under investigation, with the remaining reactions simply proceeding towards the MAC, a worthy subject of investigation but not that of the paper.
 To recreate Figure 2 in simulation, a "Custom Task", titled C3bBb Distribution Script was created in Simbiology. This script runs a series of stochastic simulations, from which the distribution of the C3bBb is sampled at several time periods. This data from each iteration is combined, and through the stochastic simulation allows for the macroscale trends that are viewed in Figure 2 to be seen.
Parameters
Figure 4: Initial Protein Concetrations Figure 5: Initial Protein Concetrations
 In the Simbiology model, the parameters of importance were those of the reaction rates and the protein concentrations in the serum. The protein concentrations were available^{[2]} and are listed in Figure 4. However, this study was only able to find several reaction rates. The remaining reaction rates were therefore chosen to be of approximately a similar order of magnitude. It is believed that this is a valid assumption because it was found that the unknown parameters could vary significantly without greatly affecting the overall reaction characteristics. The reaction rates used are shown in Figure 5.
Results
 The simulation was performed in Simbiology with 10,000 data points, each of which was sampled at the indicated points in time. Several different configurations were attempted in order to best match the results in the paper:
 The experimental configuration that produced results most closely matching those obtained experimentally was that which had only the initial concentration of Complement proteins with no renewal. This produced the results shown below:
Figure 6: Results of Alternative Pathway Simulation with only initial protein concentrations. X axis is percent coverage of pathogen, Y axis is number of pathogens. 10000 pathogens were used in this simulation.
Analysis
 The results of the simulation validate the model that biologists have created. By having the results so clearly match those obtained experimentally, we have shown that the model accurately represents the mechanisms used in the body to attack foreign particles, a model of which has until now only been postulated but never tested.
 The several different configurations that were examined allows for the model that is 'most true to life' to be chosen. This model has been found to be the one in which complement proteins are not continually introduced into the system.
 The configuration in which proteins are continually introduced caused the reaction to proceed on a quicker time scale and didn't match the experimental results.
 The disabling of the disassociation (formation of iC3b) was found to not noticeably affect the system. This suggests that the disassociation isn't an important component of the reaction to attack pathogens. Instead, it suggests that it is an important part of regulation, in which it can serve to remove excess C3b in the system when an attack is not taking place.
 Having a constant amount of C3 available quickened the reaction, although not to the same degree as continually refreshing all proteins. It thus becomes apparent that multiple elements of the reaction can restrict the overall reaction.
 These results indicate that the reaction doesn't have unlimited or continually renewing amounts of proteins. While this may seem trivial, this indicates that the reaction can occur without input from the rest of the body since it can use up the local proteins to proceed with the reaction. In fact, since adding additional proteins produces results inconsistent with those obtained experimentally, this suggests that the reactions are indeed limited locally and their timing may be an important component of regulation and recognition. Furthermore, the behaviour when boosting the primary component of the reaction, C3, indicates that the components in the reaction are carefully balanced such that adding or removing any component can have a drastic affect on the overall reaction. This indicates that the reaction is tuned to allow for a great deal of control on the part of the body.
Analytical Mathematical Model
From a purely analytical point of view, we can model the accumulation of C3bBb proteins on pathogens using a homogeneous Poisson process. To do this, we consider a timeline for each individual pathogen, starting at some time when no C3b proteins have attached to any of the pathogens. When a C3b attaches to a pathogen, we mark the time along that pathogen's timeline. Since the fluorescence of the pathogen cells is proportional to the number of attached C3b, we can take a "snapshot" of the system at any time by counting the number of marks along each timeline and plotting the frequency of the counted data. This counted data frequency plot should resemble the observed fluorescence frequency plot at that point in time.
Assumptions & Parameters
 In order to be a homogeneous Poisson process, we assume that the time periods between the arrivals of any two consecutive C3b proteins, called the interarrival times, are all independent random variables and are identically distribution according to the same Exponential distribution with rate parameter "r".
 Under the assumptions of a homogeneous Poisson process, the frequencies of the counted data should conform to a Poisson process whose parameters is a linear function of time, as in, if N is the number of C3b on a pathogen, N has a Poisson(r t) distribution with probability density function:
 f(x) = ((r t)^x) (e^(r t)) / (x!)
 The rate parameter r can be estimated from the data. If we assume each "unit" of fluorescence is equivalent to one C3bBb unit, we can take the observed fluoresence data at some given time t, take the average of the data to compute an unbiased estimate of the Poisson parameter, and divide by t to yield an estimate for r.
Modification of Model: Logistic TimeDependent Rates
 Strictly speaking, a homogeneous Poisson process as defined does NOT accurately reflect the interaction between the C3b and the pathogens. The above assumptions imply that the rate at which C3b bonds to a pathogen remains constant, and the time periods between each interaction are each independent. However, as C3b proteins begin to bind to pathogens, the system does to things: the number of C3b proteins in the blood increase and the C3bBb proteinpathogen complexes attract more C3b proteins. Essentially, these two phenomena imply that the rate r will increase as a function of N and t.
 Fortunately, we can actually add this to our model by simplifying N to be a logistic function of t. We choose the logistic function for a number of reasons:
 We would expect the number of C3b on a pathogen to grow exponentially immediately after infection because more C3b will attracted to the pathogen as the number of attached C3b increases, creating a seemingly dN/dt = a N relation.
 There should be a carrying capacity on the number of C3b that the body can handle and/or the liver can produce, which would mean that the probability of a new C3b being produced and attaching to a pathogen should eventually drop to zero, and consequently, the DISTRIBUTION of the total attached C3b should approach some kind of steadystate in which no more C3b proteins are attaching to the pathogens.
 Most importantly, the logistic equation is simple with a wellknown solution set. We only need to provide parameters for the initial number of C3b, a carrying capacity, and a growth rate.
 Another project using the logistic equation is APPM4390:Modelling Components of a Lunar Life Support System.
 The idea of using a Poisson distribution to model the number of attached C3b remains sound, as we are trying to model countable data. We define N(t) to be the solution of a logistic equation with initial condition of one C3b, a carrying capacity equal to the average of the limiting distribution, and a growth rate that a person can adjust to match the observed data appropriately, and instead of making the Poisson parameter a linear function of time, we look at the distribution of Poisson(N(t)) over time.
Results
When we employ the straight homogeneous Poisson process, we obtain theoretical distribution for the frequencies that change through time. An animation of the change in the distribution appears below:
Figure 7: Poisson distribution with linear rate parameter
For earlier points in time, the distribution matches the observed behavior by having a very large spike in the frequency of lowcount data. The shape of the distribution becomes stable after time, but the mean of the distribution continues to increase without bound.
When we use the logistic timedependent rate for the Poisson process, we obtain similar plots of the theoretical distribution, which appear in the animation below:
Figure 8: Poisson distribution with logistic rate parameter
This animation matches the observed behavior in both short and longterm behavior. Again, we observed an early spike in the frequency of lowcount data, but the distribution moves away from this early spike quite quickly. It then settles around a limiting distribution, which appears to match the observed data because the observed data shows a stable distribution after 10 seconds.
Analysis
Figure 9: Distribution of fluorescence (indicating C3b concentration) on pathogens with consistent initial distribution ^{[1]}
 The homogeneous Poisson process is not necessarily the best model for the interaction of C3b and pathogens in the blood. The positive feedback loop and the fact that C3bBb attracts more C3b violates the assumptions that the interarrival times are independent, identicallydistributed Exponential random variables with a constant arrival rate. However, if we were to compare the results from the homogeneous Poisson process to the observed results WITHOUT the positive feedback loop engaged (Figure 9), the behaviors are remarkably similar. This leads us to believe that the number of C3b proteins attached to a pathogen would indeed be a random variable with a Poisson distribution, though we may not be sure that the parameter of the distribution will change linearly with time.
 On the other hand, having the Poisson parameter be a logistic function of time creates a fairly accurate representation of the observed results WITH the positive feedback. Hence, the parameter for the Poisson distribution should be a logistic function of time, and through nonlinear curvefitting algorithms (which we exclude here due to the lack of exact data), one could choose appropriate values for the all of three necessary parameters of the logistic function.
Discussion
 The use of Matlab's Simbiology (To see another example of modeling in Simbiology see APPM4390:Modelling the dynamics of a complex lifecycle parasite.) to implement the commonly accepted model of the alternative pathway has been largely successful, producing similar results and verifying the important components of the model. Specifically, the effects of the positive feedback loop has been demonstrated to have a profound effect on the model, largely governing the rate at which the pathogen is covered and matching the expectation that this aspect was the primary component of amplification.
 With the model verified, it is then possible to investigate the consequences of varying a variety of parameters in the system. This has shown that the disassociation of C3b is not an important reaction when a pathogen is attacked. This reinforces the idea that this reaction is primarily used in a regulatory context, and furthermore that this regulatory mechanism does not serve to aid the pathogen in any meaningful way.
 The ability to vary the concentrations of the various proteins involved in the initialization of the reaction has allowed for the configuration that is most true to life to be chosen. This configuration has been found to be that with limited amounts of the proteins, indicating that there is no resupply. This in turn indicates that the system is neither a well mixed system or that the reaction is reinforced by the body, since either of these would replenish the availability of the proteins. Even if these proteins are added at a rapid rate, they have no large effect on the reaction until there is an infinite amount of each.
 The desire to model the formation of C3bBb at a larger scale led to the formation of a model of a Poisson process. While this model does not reflect the reaction as directly as that of the stochastic simulation, the underlying assumptions of a Poisson process, that of independent events occurring with an exponential distribution, mirrors what we would expect with the initial deposition of the C3bBb. The results of this model match closely with the experimental model at the beginning of the reaction. However, they differ greatly as the reaction progresses. This lends further credibility to the claim of the positive feedback being a primary component of the reaction since the Poisson process innately can represent the overall reaction with the exception of the positive feedback.
 The Poisson process correctly models the experimental results when the pathogen begins with a uniform initial distribution, alleviating many of the effects of the positive feedback loop.
Conclusion
 The accepted model for the Alternative Pathway has been validated through stochastic simulation. Many of the parameters of the system have been either investigated or tweaked towards the results obtained experimentally, producing a model that can be used in further investigations of the pathway. Furthermore, study of the macrolevel distribution has led to an accurate representation of the overall system with Poisson processes. This model demonstrates the behavior of the system with positive feedback removed and could also in the future be used to model the entire system through logistic parameter choices.
References
 ↑ ^{1.0} ^{1.1} ^{1.2} Pangburn, Michael K. & MullerEberhard, Micheal J. "The Alternative Pathway of Complement". Spring Seminars in Immunopathology. 163192
 ↑ Morley, Bernard J. & Walport, Mark J (2000). The Complement Facts Book p. 88. Academic Press. ISBN 0127333606.
