Student(s):
Matanya Horowitz and Toni Klopfenstein
Dates of Involvement:
Spring 2009 - Present
Faculty Advisor(s):
David Bortz
Graduate Mentor:
Modeling the Immune Complement Response
Summary:
Research on the part of the biomedical community has led to the creation of a model describing the Alternative Pathway of Complement. This model qualitatively predicts a set of behaviors 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 in Matlab's Simbiology, using parameters gleaned from "The Alternative Pathway of Complement", and to perform a series of stochastic simulations to verify the model, as well as periodic behavior analysis.
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.
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. Muller-Eberhard, 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 andC3a. 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.
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 that's created by the MAC allows for free diffusion to occur from between the inside and the outside of the pathogen.
Biological Phenomenon:
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 concentration crosses a threshold sufficient to begin the formation of the MAC complex. It is this process of formation that is examined in the Pangburn and Muller-Eberhard, (1984), 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, occurring 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, the immune system attempts 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:
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 3 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 macro-scale trends that are viewed in Figure 2 to be seen.
Model Expansion:
In order to more accurately model the complement system, the model was expanded to include all the intermediate steps within the reaction chains. This expanded model can be seen below in figure 4.
Figure 4: Expanded Complement Model
Because of the complexity of the system, traditional analysis is difficult and time consuming. In order to more easily view changes in the quantity of a give parameter or the net flux through the system, a script was written to vary the size of each reaction node, depending on which parameter was being examined. Thus, the entire system’s behavior was able to be viewed simultaneously in a clear, concise manner. An example image of this is below.
Figure 5: Frame of Reaction Nodes Varying in Size
Periodic Behavior:
In order to analyze changes in the system because of periodic behavior for any of the involved proteins or pathogens, the model was modified to demonstrate periodic input of the pathogen, approximated by the equation x(t)=sin(2πtA)
where x is the pathogen, and A is the period of the input. Because of a failure in the SimBiology desktop, no direct way exists for programming in an actual sinusoid, so the forcing function was introduced into the system by using a predator-prey species model that demonstrates periodicity, and using the predators as the forcing input for the pathogen. The modified model is shown below in figure 6.
A script was then created in order to analyze the data output from the modified model by using the Fast Fourier Transform (FFT) in order to determine frequencies of the behavior of the pathogen. An example of the results is shown below in Figure 7.
Figure 7: Fast Fourier Transform of Periodic Pathogen Behavior
This script is currently being modified to analyze multiple parameters in order to trace how signals may propagate through the Complement system. The output of C3 Convertase was also analyzed through the FFT script, though no obvious signals were present, so the derivative of the data curve was run through the FFT script, which may be necessary to do for other protein parameters in the system. Clearly displaying the difference between an analysis of the actual data and the analysis of the derivative of data is being examined and will be incorporated into the system.
Parameters:
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 and are listed in Figure 8. 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 9.
Figure 8: Initial Protein Concentrations Figure 9: Chemical Reaction Rates
Project Goals:
Matanya Horowitz's work includes optimizing the Complement Model in order that it accurately reflects and predicts results obtained experimentally. As well, his work involves studying the model and determining what aspects of the model are most sensitive to perturbations, revealing key parts of the pathway that could be targeted for further study and eventual modification.
Toni Klopfenstein’s work includes developing tools for investigating the periodic behavior of the concentrations of the various parameters in the system, as well as the net flux of energy through the system.
Both students plan to continue work on analyzing the immune complement system in the spring in order to continue expanding and improving analysis tools for this complex system.
About Matanya Horowitz and Toni Klopfenstein:
Matanya Horowitz is a senior at CU Boulder working concurrently on his Master’s degree. His research interests lie in the intersection between systems biology, robotics, and control systems. A current focus for his work is in research with Dr. David Bortz and Dr. John Younger in the study of Immune Complement, an important component of the body’s immune system. He has previously been involved in robotics research with Dr. Todd Murphey and hopes to further his interests next year as he pursues a PhD in one of the aforementioned fields.
Toni Klopfenstein is currently a fourth-year Applied Mathematics major, with an emphasis in Integrative Physiology and a minor in Italian. She works as a tutor in the engineering college at CU, and organizes K-12 outreach events in order to teach younger students about the wonders of mathematics and engineering. She hopes to continue within the field of medicine after graduation.
References:
1. Pangburn, Michael K. & Muller-Eberhard, Micheal J. "The Alternative Pathway of Complement". Spring Seminars in Immunopathology. 163-192
2. Morley, Bernard J. & Walport, Mark J (2000). The Complement Facts Book p. 88. Academic Press. ISBN 0-12-733360-6.
3. Ablowitz, M. and Fokas, A. Complex Variables. Cambridge University Press. New York, 2nd Edition, 2003.
4. Haberman, R. Applied Partial Differential Equations. Pearson Prentice Hall. New Jersey, 4th Edition, 2004.
