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.