Characterizing the Design Space of Oscillatory Biological Networks

Strom Truman Clark strom.clark@gmail.com stromtc

Dr. Mark Transtrum, Department of Physics and Astronomy

### Introduction

Characterizing the relevant parameters of a design space in order to satisfy a

specific behavior criterion is an important problem throughout all of science and

engineering. In this project we proposed to apply model reduction to the case of

biological oscillations involving Michaelis-Menten reactions. By removing irrelevant

parameters from a fully connected network we were able to reduce a known problem in

systems biology to a more general model. Furthermore, significant progress has been

made in applying Manifold Boundary Approximation Method (MBAM) to oscillatory

models in systems biology.

### Methodology

An important problem in science is how to characterize the relevant physics that

controls a particular behavior. For example, many biological systems are oscillatory,

and the properties of these oscillations are critical to the fitness of the organism. A

controlled experiment may identify some control knob in the system that alters several

features of the oscillations. Many further experiments identify other control knobs that

tune other features. From this information, we are able to identify the relevant control

knob for single feature. That is to say, we seek a characterization of the space that

simultaneously allows one to control individual components of behavior and

exhaustively enumerates all such ways that behavior can be altered. The problem

becomes exponentially difficult as the number of parameters increases.

To address this problem we applied MBAM to models in systems biology with

known behaviors and fully-connected networks of arbitrary biological systems. We are

able to identify the relevant parameters by removing the parameter combinations that

have no influence on the system behavior. In a fully-connected system each node is

connected in every possible way (i.e., each node has a positive and negative regulatory

connection to every other node as well as positive self-regulation and environmental

inhibitions). The resulting model is “sloppy,” meaning that it is controlled by only a

handful of parameter combinations. These “stiff” parameter combinations are the

relevant control knobs that we seek to isolate through model reduction. We can then

identify analogous control knobs in other real systems or design synthetic systems that

exhibit a desired behavior.

Mathematical models can be seen as manifolds which are N dimensional shapes

embedded in N dimensional space, where N is the the number of parameters in the

model. The edges of this model manifold are an approximation of the model, but with

one less parameter. We used information theory and statistical methods including the

Fisher Information Matrix to identify relevant parameter combinations and MBAM to

iteratively remove unnecessary parameters. This iterative example is explained below:

First we identify some behavior to be fit. We then fit the parameters to match the

desired behavior. Then, using the Fisher Information Matrix (FIM), we identify the most

“sloppy” parameter, which is generally indicated by the smallest eigenvalue of the FIM.

This eigenvalue has a corresponding direction in the manifold which direction leads to

an edge of the manifold. We can identify this edge by heading along the identified

eigenvector on what is called a geodesic. The geodesic is an analog to a straight line in

curved space and is numerically constructed by solving a set of differential equations

using the methods of computational differential geometry. As the boundary of the

manifold is approached along the geodesic, the parameters exhibit a limiting behavior.

By identifying and solving these mathematical limits, which reduces the dimensionality

of the model by exactly one, we identify the boundary of the manifold which is an

approximate model with one less parameter. Finally we use the parameter values at

the end of the geodesic to fit the parameters for the next iteration.

### Results

As a proof of principle, we began by applying MBAM to the repressilator a

famous oscillatory problem in systems biology. This is a synthetic system of oscillations

in which gene concentrations shift in bacteria. While technically a gene network, We

showed that modeling the system as a protein network using Michaelis-Menten kinetics

mimicked the behavior of the repressilator in the literature. Through applying MBAM, we

also showed that the repressilator gene network can be reduced from twelve

parameters to a simple harmonic oscillator having only three parameters.

In applying MBAM to a fully connected network of three nodes containing

thirty-six edges or parameters, we opted to first find many fits to which we would apply

methods of automated reduction to find reduced models. We were ultimately able to

reduce the parameters of several of the fits by almost half.

When we applied MBAM to fully connected networks, the limits were much more

difficult to identify and evaluate. While many limits involving one parameter going to

infinity or zero were automated, often the correct limit would at first glance appear to be

wrong. In essence, while in theory the methods for identifying the best eigendirection

are straightforward, evaluating the limit to determine if the correct direction was in-deed

chosen is often much more difficult. We believe the resulting models can be further

reduced; however, much more footwork is required to meet the difficult task of

evaluating diverse mathematical limits.

### Discussion and Conclusion

Oscillatory behavior is critical for the proper functioning of many biological

processes, such as circadian rhythms. By isolating parameter combinations in networks

of chemical reactions that are responsible for controlling a particular function, we seek

to provide engineers and experimentalists with the information to design a desired

behavior or fix an errant phenotype. More broadly, by identifying which parameters

affect oscillatory behaviors we will gain a more general understanding of how

interactions among specific biological components control phenotypes in real

organisms.

Additional research by others in the Transtrum lab has shown that MBAM can be

successfully applied to adaptation in biological systems. The reduced models for the

fully connected network found during the course of this project are likely not minimal,

but do show that MBAM can be applied to the problem to yield reduced models with

roughly half the parameters. We believe MBAM can be further applied to find minimal

oscillatory models.