# <span class=strong>Reception and Poster Session</span>

Monday, May 12, 2008 - 5:00pm - 6:30pm

Lind 400

**Stochastic models for intracellular reaction networks**

Yiannis Kaznessis (University of Minnesota, Twin Cities)

In the presented work we will describe how to rationalize synthetic biology

using model-driven, molecular-level engineering principles. In the poster

we will focus on the theoretical effort to develop an algorithm for

simulating biomolecular systems across all relevant time and length scales;

from stochastic-discrete to stochastic-continuous and

deterministic-continuous models, we are developing the theoretical

foundation for accurately simulating all biomolecular interactions in

transcription, translation, regulation and induction and how these result in

phenotypic probability distributions at the population level.**Interlaced Euler scheme for stiff stochastic differential equations**

Ioana Cipcigan (University of Maryland Baltimore County)Muruhan Rathinam (University of Maryland Baltimore County)

In deterministic as well as stochastic models of chemical kinetics,

stiff systems, i.e. systems with vastly different time scales where the

fast scales are stable, are very common. It is well known that

the implicit Euler method is well suited for stiff deterministic

equations (modeled by ODEs) while the explicit Euler is not. In particular

once the fast transients are over, the implicit Euler allows for the choice of times steps comparable to the slowest time scale of the system.

In stochastic systems (modeled by SDEs) the picture is more complex.

While the implicit Euler has better stability properties over the

explicit Euler, it underestimates the stationary variance. In general one

may not expect any method to work successfully by taking time steps of the

order of the slowest time scale. We explore the idea of interlacing large

implicit Euler steps with a sequence of small explicit Euler steps.

In particular we present our study of a linear test system of SDEs and

demonstrate that such interlacing could effectively deal with

stiffness. We also show uniform convergence of mean and variance.**Stochastic simulations of reaction-diffusion models**

Anilkumar Devarapu (University of Louisville)

Reaction-Diffusion models are key components of models in developmental biology.

These reaction diffusion processes can be mathematically modelled using either deterministic partial differential equations or

stochastic simulation algorithms. Here we discuss the stochastic simulations on both linear and non-linear Reaction-diffusion models**Solutions to inverse problems of biochemical networks using stochastic methods**

Mónica Bugallo (The State University of New York)Petar Djuric

Advances in the development of models that can satisfactorily describe biochemical networks are extremely valuable for understanding life processes. In order to get full description of such networks, one has to solve the inverse problem, that is, estimate unknowns (rates and populations of various species) or choose models from a set of hypothesized models using experimental data. In this work we discuss signal processing techniques for resolving the inverse problem of biochemical networks using a stochastic approach based on Bayesian theory. The proposed methods are tested in simple scenarios and the results are promising and suggest application of these methods to more complex networks.**Product form stationary distributions for deficiency zero chemical reaction networks**

David Anderson (University of Wisconsin, Madison)

The dynamics of chemical reaction networks can be modeled either deterministically or stochastically. The deficiency zero theorem for deterministically modeled systems gives conditions under which a unique equilibrium value with strictly positive components exists within each stoichiometric compatibility class (linear subset of Euclidean space in which trajectories are bounded). The conditions of the theorem actually imply the stronger result that there exist concentrations for which the network is complex balanced. That observation in turn implies that the standard stochastic model for the reaction network has a product form stationary distribution.**Multiscale methods in a heat shock response model**

Hye-Won Kang (University of Wisconsin, Madison)

We analyze a heat shock response model developed by Srivastava, Peterson, and Bently (2001) to find appropriate scaling for the abundance of molecules of chemical species and for chemical reaction rates. Extending the method introduced in the paper by Ball, Kurtz, Popovic, and Rempala (2006), we apply the multiple scaling method to the various range of the abundance of chemical species and the chemical reaction rate constants.

In this method, the complicated original system is split into several subsystems with separate fast, intermediate, and slow processes by using the approximated scaled processes with appropriate time change. Then, we can approximate the state of the system from the state of the reduced subsystems. In the slow time scale, we identify behavior of fast processes since they already start to move while the slow species remain mostly constant. In the fast time scale, we identify behavior of the slow processes driven by averaged behavior of fast processes.

Our goal is to find an appropriate scaling that makes the abundance of molecules of each chemical species well-balanced. To make each chemical species balanced, the maximal order of magnitude of the reaction rates of production for each chemical species must be the same as the maximal order of magnitude of the reaction rates of consumption. We also need a balance condition between the maximal order of magnitude of collective reaction rates of inflow and outflow in each atom graph. In case any of balance conditions are not satisfied, we put restriction on the range of time change exponent. In this poster, I will introduce multiscale method briefly and give a simulation result for one set of

exponents meeting our balance conditions.**Stochastic control analysis for biological reaction systems**

Kyung Kim

We have investigated how noise propagation in biological reaction networks affects system sensitivities. We have shown that the sensitivities can be enhanced in one region of system parameter values by reducing sensitivities in another region. We have applied this sensitivity compensation effect to enhance the amplification of a concentration detector designed by using an incoherent feed-forward reaction network. We have also shown that metabolic control analysis can be extended for stochastic reaction systems by providing new versions of the summation and connectivity theorems.**Stochastic modeling of vesicular stomatitis virus(VSV)**

growth in cells: An application of order statistics to predict

replication delays

Rishi Srivastava (University of Wisconsin, Madison)

Although viruses are the smallest organisms with the shortest genomes, they have major impacts on human health, causing deadly diseases (e.g., influenza, AIDS, cancer) on a global scale. To reproduce, a virus particle must infect a living cell and divert biosynthetic resources toward production of virus components. A better mechanistic understanding the infection cycle could lead to insights for more effective anti-viral strategies. However, simulating the virus infection cycle is a computationally hard problem because some species fluctuate rapidly while others change gradually in number. We study vesicular stomatitis virus, an experimentally accessible virus for which we have developed a deterministic kinetic model of growth (Lim et al, PLoS Comp Bio, 2006). Stochastic simulation of VSV genome encapsidation is a computationally-intensive chain reaction that produces rapid fluctuations in the nucleocapsid(N) protein that associates with the genome. Analytical results from order statistics enable us to avoid explicit tracking of all intermediate species, with significant reduction in computational burden. This approach can be generalized to a broad class of stochastic polymerization reactions where multi-step chain reactions are modeled as a single reaction with time-delay.**Predicting translation rate from sequence**

Howard Salis (University of California, San Francisco)

The reliable genetic engineering of bacterial systems would be

greatly aided by a more quantitative and predictive understanding of

gene expression. In many synthetic biology applications, the

production rate of a protein needs to be precisely tuned to optimize

some performance characteristic of the genetic system. A typical

application might require selecting the optimal expression level of

enzymes in a metabolic pathway. The production rate of a protein can

be precisely tuned by varying the mRNA sequence of the 5' UTR

(untranslated region), including the ribosome binding site (RBS), that

precedes the protein's coding sequence. However, there is currently no

quantitative model that can predict an RNA sequence that yields a

desired production rate of protein.

We have developed and experimentally validated a statistical

thermodynamic model of translation initiation in Escherichia coli. The

model quantifies the effects of the important RNA, RNA-RNA, and

RNA-protein interactions and determines the resulting translation

initiation rate of a specified mRNA transcript. We combine the

quantitative model with Monte Carlo sampling to create a design method

that generates synthetic 5' UTRs. The user inputs a protein coding

sequence and a desired translation initiation rate and the design

method will generate the corresponding RNA sequence. We measure the

accuracy of the design method by generating numerous synthetic 5' UTRs

that drive the expression of the fluorescent protein RFP in a

simplified test system. We then compare the design method's

predictions with flow cytometry data from a test system. The design

method is capable of accurately generating RNA sequences that yield

the desired translation rate, with rates varying across three orders

of magnitude.

The presented model and method is a key foundational technology that

will enable us to more rationally engineer synthetic biological

systems. With the emergence of lower cost large-scale DNA synthesis

and the rapid assembly of genetic systems, the development bottleneck

now shifts towards identifying the DNA sequence that ultimately yields

a desired system behavior. These and other quantitative and predictive

models aim to provide that much-needed missing link.**Biochemical and network modeling of the mammalian suprachiasmatic nucleus**

Richard Yamada (University of Michigan)

The Supra-Chiasmatic Nucleus (SCN) is the central oscillator that keeps time in all mammalian organisms. Central to understanding the circadian rhythms that these oscillators generate are the multitude of interacting genes and proteins, in addition to the complex interactions of coupled SCN cells.

In recent year, quantitative modelling has emerged as an additional tool complimenting experimental techniques in the study of circadian rhythms. In this poster, we discuss a detailed mathematical model for circadian timekeeping within the SCN. Our proposed model consists of a large population of SCN neurons, with each neuron containing a network of biochemical reactions involving the core circadian components. Central to our work is the determination of the model’s unkown parameters, which were obtained from comparing the model’s output to experimental data. From these estimated parameters, additional experimental test of the model are proposed. Our studies highlight the importance of low numbers of molecules of clock proteins, and how this fact addects the accuracy of circadian timekeeping.**Stochastic modeling of bistable chemical systems:**

Schlogl's model

Melissa Vellela (University of Washington)

When describing chemical systems in a nonequilibrium steady state, striking differences arise in the predictions between deterministic and stochastic models. Using a classic model first introduced by Schlogl, we study the steady state behaviors of a one dimensional, open system that exhibits bistability. The definitions of thermodynamic quantities, such as flux, chemical potential, and entropy production rate are compared across the two types of models. The stochastic model allows for jumping between the two stable steady states which are separated by an unstable steady state in the deterministic model. The transition rates and exponentially small eigenvalue associated with this jumping are investigated and calculated numerically. Understanding the behavior of the stochastic steady state directly from the master equation is important because even widely used approximations such as Fokker-Planck can be incorrect, as is shown for this example.**Simulating discrete biochemical reaction systems**

Arnab Ganguly (University of Wisconsin, Madison)

The stochastic simulation algorithm, or Gillespie Algorithm, is a tool used to simulate discrete biochemical reaction systems when there are a small to moderate number of molecules. The Gillespie Algorithm has been adapted to handle systems with delays (such as with gene transcription and translation) and approximations (such as tau leaping) have been developed in order to increase computation speed. In this poster we will model discrete biochemical systems via a random

time change representation. We will then demonstrate how natural and

efficient modifications of the stochastic simulation algorithm arise

from such a representation. Also, we will use this representation to

show how to incorporate post-leap checks in the tau-leaping algorithm.