DNA as a universal substrate for chemical kinetics

Molecular programming aims to systematically engineer molecular and chemical systems of autonomous function and ever-increasing complexity. A key goal is to develop embedded control circuitry within a chemical system to direct molecular events. Here we show that systems of DNA molecules can be constructed that closely approximate the dynamic behavior of arbitrary systems of coupled chemical reactions. By using strand displacement reactions as a primitive, we construct reaction cascades with effectively unimolecular and bimolecular kinetics. Our construction allows individual reactions to be coupled in arbitrary ways such that reactants can participate in multiple reactions simultaneously, reproducing the desired dynamical properties. Thus arbitrary systems of chemical equations can be compiled into real chemical systems. We illustrate our method on the Lotka–Volterra oscillator, a limit-cycle oscillator, a chaotic system, and systems implementing feedback digital logic and algorithmic behavior

Recursively constructing analytic expressions for equilibrium distributions of stochastic biochemical reaction networks

Noise is often indispensable to key cellular activities, such as gene expression, necessitating the use of stochastic models to capture its dynamics. The chemical master equation (CME) is a commonly used stochastic model of Kolmogorov forward equations that describe how the probability distribution of a chemically reacting system varies with time. Finding analytic solutions to the CME can have benefits, such as expediting simulations of multiscale biochemical reaction networks and aiding the design of distributional responses. However, analytic solutions are rarely known. A recent method of computing analytic stationary solutions relies on gluing simple state spaces together recursively at one or two states. We explore the capabilities of this method and introduce algorithms to derive analytic stationary solutions to the CME. We first formally characterize state spaces that can be constructed by performing single-state gluing of paths, cycles or both sequentially. We then study stochastic biochemical reaction networks that consist of reversible, elementary reactions with two-dimensional state spaces. We also discuss extending the method to infinite state spaces and designing the stationary behaviour of stochastic biochemical reaction networks. Finally, we illustrate the aforementioned ideas using examples that include two interconnected transcriptional components and biochemical reactions with two-dimensional state spaces

Stochastic Chemical Reaction Networks for Robustly Approximating Arbitrary Probability Distributions

We show that discrete distributions on the d-dimensional non-negative integer lattice can be approximated arbitrarily well via the marginals of stationary distributions for various classes of stochastic chemical reaction networks. We begin by providing a class of detailed balanced networks and prove that they can approximate any discrete distribution to any desired accuracy. However, these detailed balanced constructions rely on the ability to initialize a system precisely, and are therefore susceptible to perturbations in the initial conditions. We therefore provide another construction based on the ability to approximate point mass distributions and prove that this construction is capable of approximating arbitrary discrete distributions for any choice of initial condition. In particular, the developed models are ergodic, so their limit distributions are robust to a finite number of perturbations over time in the counts of molecules

Correcting Errors Due to Species Correlations in the Marginal Probability Density Evolution Algorithm

Synthetic biology is an emerging field that integrates and applies engineering design methods to biological systems. Its aim is to make biology an engineerable science. Over the years, biologists and engineers alike have abstracted biological systems into functional models that behave similarly to electric circuits, thus the creation of the subfield of genetic circuits. Mathematical models have been devised to simulate the behavior of genetic circuits in silico. Most models can be classified into deterministic and stochastic models. The work in this dissertation is for stochastic models. Although ordinary differential equation (ODE) models are generally amenable to simu- late genetic circuits, they wrongly assume that a system\u27s chemical species vary continuously and deterministically, thus making erroneous predictions when applied to highly stochastic systems. Stochastic methods have been created to take into account the variability, un- predictability, and discrete nature of molecular populations. The most popular stochastic method is the stochastic simulation algorithm (SSA). These methods provide a single path of the overall pool of possible system\u27s behavior. A common practice is to take several inde- pendent SSA simulations and take the average of the aggregate. This approach can perform iv well in low noise systems. However, it produces incorrect results when applied to networks that can take multiple modes or that are highly stochastic. Incremental SSA or iSSA is a set of algorithms that have been created to obtain ag- gregate information from multiple SSA runs. The marginal probability density evolution (MPDE) algorithm is a subset of iSSA which seeks to reveal the most likely qualitative behavior of a genetic circuit by providing a marginal probability function or statistical enve- lope for every species in the system, under the appropriate conditions. MPDE assumes that species are statistically independent given the rest of the system. This assumption is satisfied by some systems. However, most of the interesting biological systems, both synthetic and in nature, have correlated species forming conservation laws. Species correlation imposes con- straints in the system that are broken by MPDE. This work seeks to devise a mathematical method and algorithm to correct conservation constraints errors in MPDE. Furthermore, it aims to identify these constraints a priori and efficiently deliver a trustworthy result faithful to the true behavior of the system

Stochastic Chemical Reaction Networks for Robustly Approximating Arbitrary Probability Distributions

We show that discrete distributions on the d-dimensional non-negative integer lattice can be approximated arbitrarily well via the marginals of stationary distributions for various classes of stochastic chemical reaction networks. We begin by providing a class of detailed balanced networks and prove that they can approximate any discrete distribution to any desired accuracy. However, these detailed balanced constructions rely on the ability to initialize a system precisely, and are therefore susceptible to perturbations in the initial conditions. We therefore provide another construction based on the ability to approximate point mass distributions and prove that this construction is capable of approximating arbitrary discrete distributions for any choice of initial condition. In particular, the developed models are ergodic, so their limit distributions are robust to a finite number of perturbations over time in the counts of molecules