Phase Reduction of Stochastic Biochemical Oscillators

Formal methods have enabled breakthroughs in many fields, such as in hardware verification, machine learning and biological systems. The key object of interest in systems biology, synthetic biology, and molecular programming is chemical reaction networks (CRNs) which formalizes coupled chemical reactions in a well-mixed solution. CRNs are pivotal for our understanding of biological regulatory and metabolic networks, as well as for programming engineered molecular behavior. Although it is clear that small CRNs are capable of complex dynamics and computational behavior, it remains difficult to explore the space of CRNs in search for desired functionality. We use Alloy, a tool for expressing structural constraints and behavior in software systems, to enumerate CRNs with declaratively specified properties. We show how this framework can enumerate CRNs with a variety of structural constraints including biologically motivated catalytic networks and metabolic networks, and seesaw networks motivated by DNA nanotechnology. We also use the framework to explore analog function computation in rate-independent CRNs. By computing the desired output value with stoichiometry rather than with reaction rates (in the sense that X → Y+Y computes multiplication by 2), such CRNs are completely robust to the choice of reaction rates or rate law. We find the smallest CRNs computing the max, minmax, abs and ReLU (rectified linear unit) functions in a natural subclass of rate-independent CRNs where rate-independence follows from structural network properties.

The statistics of the condensed polaritons is described in terms of the Wigner function. In the framework of the truncated Wigner method, the Wigner function obeys a Fokker- Planck equation, which is solved analytically. The second order correlations in the stationary state are in excellent agreement with those obtained from the numerical solution of the master equation and show a qualitative and, well above threshold, also quantitative agreement with recent experiments. Furthermore, the contributions of the different noise effects that influence the polariton ground state statistics are explicitly defined. Exploiting the equivalence between Fokker-Planck and Langevin descriptions of stochastic processes, the time dependent correlations of the polaritons close to the stationary state are derived. Explicit expressions for the linewidth and for the relaxation rate of the polariton intensity are obtained, whose values are of the same order of magnitude of the experimental data. Finally, the limit of validity of the truncated Wigner method in the present model is discussed.

In molecular programming, the Chemical Reaction Network model is often used to describe systems of interacting molecules. This model can describe either real systems, allowing us to analyze and determine their computational function; or describe hypothetical systems, with known computational function but perhaps no known physical example. One significant breakthrough in the field is that any Chemical Reaction Network can be approximated by a system using DNA Strand Displacement mechanisms. This allows the Chemical Reaction Network model to be treated like a programming language, where programs can be written in the abstract and then compiled into physical molecules. Given a programming language and a proof-of-concept compiler, one would want to take the compiler from the proof-of-concept stage into a more reliable, more systematic, and better understood process. This thesis is made up of my contributions to that effort. First, given a programming language and a compiler, it would be useful to formally verify that the compiler is correct. My collaborators, Qing Dong and Erik Winfree, and I defined a Chemical Reaction Network-specific form of bisimulation equivalence, which can compare two such networks and verify that one is (or is not) a correct implementation of the other. For example, the compiler-produced DNA circuit can be verified as an implementation of its abstract program, although this is not the only possible use. After defining this concept of equivalence, we show that it can be checked by algorithm; although various parts of the problem are NP-complete or PSPACE-complete, we give algorithms that meet these lower bounds. We also prove a number of interesting properties of Chemical Reaction Network bisimulation equivalence, including transitivity and modularity properties which are particularly useful for stepwise checking of large systems. We further extend this bisimulation method to linear Polymer Reaction Networks, a strictly more powerful abstraction which has been occasionally used in molecular programming. Again we prove complexity hardness results, which in this case are as expected uncomputable in the general case; however, many practical systems can still be verified, and we give one such example. Finally, we use bisimulation to identify a class of single-locus networks that are practical to implement. Thus we show a method of verification which can simplify use of the above-mentioned compiler by proving general statements of correctness about its results. Second, given a programming language and a concept of compiling it, it would be useful to optimize the result of the compilation. One particular area of optimization is the number of DNA strands per prepared complex; some experiments suggest that systems with no more than 2 strands per complex are more robust. Lulu Qian and I developed some proposed DNA Strand Displacement schemes for general Chemical Reaction Network implementations with no more than 2 strands per complex, and a number of other desirable properties. Meanwhile, having been shown to be useful for many reasons, the mechanisms of DNA Strand Displacement have recently been formalized, abstracted, and analyzed. I show that this formalization, combined with the bisimulation methods above, can prove various statements about the limits of DNA Strand Displacement systems. For example, a set of desirable conditions including the 2-strand limit cannot be achieved by any general Chemical Reaction Network implementation scheme. I also observe that two of the new schemes we discovered, each meeting all but one condition of the impossible set, were found in the process of coming up with this proof. I thus argue that through formalization of DNA Strand Displacement we can have a more systematic method of finding and designing molecular programs, and of knowing when the programs we want do not exist.

Our main focus is on a general class of active rotators with mean field interactions, that is, globally coupled large families of dynamical systems on the unit circle with nontrivial stochastic dynamics. The dynamics of each isolated system is ${\mathrm d} \psi_t =-\delta V'(\psi_t){\mathrm d} t + {\mathrm d} w_t$, where $V'$ is a periodic function, $w$ is a Brownian motion, and $\delta$ is an intensity parameter. It is well known that the interacting dynamics is accurately described, in the limit of infinitely many interacting components, by a Fokker--Planck PDE and the model reduces for $\delta=0$ to a particular case of the Kuramoto synchronization model, for which one can show the existence of a stable normally hyperbolic manifold of stationary solutions for the corresponding Fokker--Planck equation. (We are interested in the case in which this manifold is nontrivial, which happens when the interaction is sufficiently strong, that is, in the synchronized regime of the Kuramoto model.) We use the robustness of normally hyperbolic structures to infer qualitative and quantitative results on the $\vert\delta\vert \le \delta_0$ cases with $\delta_0$ a suitable threshold; in fact, we obtain an accurate description of the dynamics on the invariant manifold for $\delta \neq0$ and we link it explicitly to the potential $V$. This approach allows us to have a complete description of the phase diagram of the active rotator model, at least for $\vert \delta\vert \le \delta_0$, thus identifying for which values of the parameters (notably, noise intensity and/or coupling strength) the system exhibits periodic pulse waves or stabilizes at a quiescent resting state. Moreover, some of our results are very explicit and this brings a new insight into the combined effect of active rotator dynamics, noise, and interaction. The links with the literature on specific systems, notably neuronal models, are discussed in detail.

Quantifying effects of stochastic focusing and defocusing on noise propagation in biochemical reaction networks

In certain classes of chemical reaction networks (CRN), there may be two stable states. The challenge is to find a model of the CRN such that the stability properties can be predicted. In this paper we consider the problem of building a P-system designed to simulate the CRN in an attempt to determine if the CRN is stable or bistable. We found that for the networks in [2] none of the bistable CRN would have a bistable P-system by stoichiometry alone. The reaction kinetics must be included in the P-system model; the implementation of which has been considered an open problem. In this paper we conclude that a P-system for a CRN in mreactants and nproducts has at most 2(m2+ mn) membranes and 6(m2+ mn) rules. This suggests that P-system models of a chemical reaction network, including both stoichiometry and reaction kinetics can be built.

Chemical Reaction Networks (CRNs) provide a useful abstraction of molecular interaction networks in which molecular structures as well as mass conservation principles are abstracted away to focus on the main dynamical properties of the network structure. In their interpretation by ordinary differential equations, we say that a CRN with distinguished input and output species computes a positive real function \(f:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}\), if for any initial concentration x of the input species, the concentration of the output molecular species stabilizes at concentration f(x). The Turing-completeness of that notion of chemical analog computation has been established by proving that any computable real function can be computed by a CRN over a finite set of molecular species. Rate-independent CRNs form a restricted class of CRNs of high practical value since they enjoy a form of absolute robustness in the sense that the result is completely independent of the reaction rates and depends solely on the input concentrations. The functions computed by rate-independent CRNs have been characterized mathematically as the set of piecewise linear functions from input species. However, this does not provide a mean to decide whether a given CRN is rate-independent. In this paper, we provide graphical conditions on the Petri Net structure of a CRN which entail the rate-independence property either for all species or for some output species. We show that in the curated part of the Biomodels repository, among the 590 reaction models tested, 2 reaction graphs were found to satisfy our rate-independence conditions for all species, 94 for some output species, among which 29 for some non-trivial output species. Our graphical conditions are based on a non-standard use of the Petri net notions of place-invariants and siphons which are computed by constraint programming techniques for efficiency reasons.

We consider how to generate chemical reaction networks (CRNs) from functional specifications. We propose a two-stage approach that combines synthesis by satisfiability modulo theories and Markov chain Monte Carlo based optimisation. First, we identify candidate CRNs that have the possibility to produce correct computations for a given finite set of inputs. We then optimise the reaction rates of each CRN using a combination of stochastic search techniques applied to the chemical master equation, simultaneously improving the of correct behaviour and ruling out spurious solutions. In addition, we use techniques from continuous time Markov chain theory to study the expected termination time for each CRN. We illustrate our approach by identifying CRNs for majority decision-making and division computation, which includes the identification of both known and unknown networks.

Stochastic chemical reaction networks (CRNs) are complex systems that combine the features of concurrent transformation of multiple variables in each elementary reaction event and nonlinear relations between states and their rates of change. Most general results concerning CRNs are limited to restricted cases where a topological characteristic known as deficiency takes a value 0 or 1, implying uniqueness and positivity of steady states and surprising, low-information forms for their associated probability distributions. Here we derive equations of motion for fluctuation moments at all orders for stochastic CRNs at general deficiency. We show, for the standard base case of proportional sampling without replacement (which underlies the mass-action rate law), that the generator of the stochastic process acts on the hierarchy of factorial moments with a finite representation. Whereas simulation of high-order moments for many-particle systems is costly, this representation reduces the solution of moment hierarchies to a complexity comparable to solving a heat equation. At steady states, moment hierarchies for finite CRNs interpolate between low-order and high-order scaling regimes, which may be approximated separately by distributions similar to those for deficiency-zero networks and connected through matched asymptotic expansions. In CRNs with multiple stable or metastable steady states, boundedness of high-order moments provides the starting condition for recursive solution downward to low-order moments, reversing the order usually used to solve moment hierarchies. A basis for a subset of network flows defined by having the same mean-regressing property as the flows in deficiency-zero networks gives the leading contribution to low-order moments in CRNs at general deficiency, in a 1/n expansion in large particle numbers. Our results give a physical picture of the different informational roles of mean-regressing and non-mean-regressing flows and clarify the dynamical meaning of deficiency not only for first-moment conditions but for all orders in fluctuations.

We study the composability of discrete chemical reaction networks (CRNs) that stably compute (i.e., with probability 0 of error) integer-valued functions $$f:\mathbb {N}^d\rightarrow \mathbb {N}$$. We consider output-oblivious CRNs in which the output species is never a reactant (input) to any reaction. The class of output-oblivious CRNs is fundamental, appearing in earlier studies of CRN computation, because it is precisely the class of CRNs that can be composed by simply renaming the output of the upstream CRN to match the input of the downstream CRN. Our main theorem precisely characterizes the functions f that are stably computable by output-oblivious CRNs with an initial leader. The key necessary condition is that for sufficiently large inputs, f is the minimum of a finite number of nondecreasing quilt-affine functions (an affine function is linear with a constant offset; a quilt-affine function is linear with a periodic offset).

Traditionally, quantitative models of reaction networks are based on classical chemical kinetics. Under the assumption of the thermodynamic limit (infinite number of molecules/volume limit), reactions are modeled as continuous, deterministic processes. It has become evident, however, that discrete fluctuations play a crucial role in cellular processes like gene expression and signal transduction, where constituents are typically present in small numbers. In this case, a modeling approach based on stochastic reaction kinetics is required, where reactions are modeled as discrete stochastic processes. The temporal evolution of the probability density function of the number of molecules is given by the chemical master equation (CME), which, however, is impractical to solve in most applications due to its high dimensionality. Instead, it is common practice to approximate an indirect solution of the CME by computing realizations of the underlying Markov jump process. A major aim is the development of such indirect approaches that enable the simulation of complex multi-scale reaction networks. This thesis deals with the promising development of hybrid methods, where fast reactions associated with large numbers of molecules are continuously and deterministically approximated, and all other reactions are modeled as discrete stochastic processes. We demonstrate the benefit of such a hybrid system description on an integrative model of the replication dynamics of the human immunodeficiency virus (HIV). Based on hybrid simulations, we are able to design and validate in silico a novel treatment strategy for HIV-infected patients that can lead to significant improvements compared to conventional treatment strategies. While current hybrid methods almost exclusively rely on indirect approximations, a novel hybrid approach is presented in this thesis that allows to solve the CME directly. Based on a multi-scale expansion, evolution equations are derived that couple a CME on a reduced state space to evolution equations of deterministically approximated variables. Thus, the impact of changes in the probability distribution of the stochastic subsystem on the dynamics of the deterministic components becomes apparent and, in contrast to indirect hybrid methods, is taken into account explicitly. We illustrate and discuss the performance of this direct hybrid approaches in application to model systems of biological interest. In the last part of this thesis, we derive effective protein synthesis rates as typically incorporated in deterministic models of biochemical systems by reduction of a detailed stochastic model of gene expression. We use this approach to derive a protein interactions model of the flagellar gene regulation cascade in Escherichia coli. Based on the deduced functional relations between transcriptional and translational processes on the one hand and the synthesis rates on the other, we find that sensitivity with respect to effective rates does not directly carry over to the aggregated subprocesses.

Noise slows the rate of Michaelis–Menten reactions

The Chemical Master Equation (CME) is a stochastic and discrete-state continuous-time model for macromolecular reaction networks inside the cell. Under this theoretical framework, the solution of a sparse linear system provides the steady-state probability landscape over the molecular microstates. The CME framework can in fact reveal important insights into basic principles on how biological networks function, having critical applications in stem cell study and cancer development. However, the exploratory nature of system biology research involves the solution of the same reaction network under different conditions. As a result, the application of the CME framework is feasible only if we are able to solve several large linear systems in a reasonable amount of time. In recent years, GPU has emerged as a cost-effective high performance architecture easily available to bioscientists around the world. In this paper, we propose an efficient GPUbased Jacobi iteration for steady-state probability calculation. We provide several optimization strategies based on the problem structure with the aim of outperforming the conventional multicore implementation while minimizing the GPU memory footprint. We combine an ELL+DIAG sparse matrix format with DFS ordering to leverage the diagonal density. Moreover, we devise an improved sliced ELL sparse matrix representation based on warp granularity and local rearrangement. Experimental results demonstrate an average double-precision performance of 14.212 GFLOPS in solving the CME (a speedup of 15.67x compared to the optimized Intel MKL library). Our implementation of the warp-grained sliced ELL format outperforms the state-of-the-art in terms of SpMV performance (a speedup of 1.24x over clSpMV). Moreover, it shows consistent improvements for a wider set of application domains and a good memory footprint reduction. The results achieved in this work provide the foundation for applying the CME framework to realistic biochemical systems. In addition, our GPU-based steady-state computation can be generalized to operation on stochastic matrices (Markov models), achieving good performance with matrix structures similar to biological reaction networks.

Reaction networks and kinetics of biochemical systems

BackgroundStochastic modeling and simulation provide powerful predictive methods for the intrinsic understanding of fundamental mechanisms in complex biochemical networks. Typically, such mathematical models involve networks of coupled jump stochastic processes with a large number of parameters that need to be suitably calibrated against experimental data. In this direction, the parameter sensitivity analysis of reaction networks is an essential mathematical and computational tool, yielding information regarding the robustness and the identifiability of model parameters. However, existing sensitivity analysis approaches such as variants of the finite difference method can have an overwhelming computational cost in models with a high-dimensional parameter space.ResultsWe develop a sensitivity analysis methodology suitable for complex stochastic reaction networks with a large number of parameters. The proposed approach is based on Information Theory methods and relies on the quantification of information loss due to parameter perturbations between time-series distributions. For this reason, we need to work on path-space, i.e., the set consisting of all stochastic trajectories, hence the proposed approach is referred to as “pathwise”. The pathwise sensitivity analysis method is realized by employing the rigorously-derived Relative Entropy Rate, which is directly computable from the propensity functions. A key aspect of the method is that an associated pathwise Fisher Information Matrix (FIM) is defined, which in turn constitutes a gradient-free approach to quantifying parameter sensitivities. The structure of the FIM turns out to be block-diagonal, revealing hidden parameter dependencies and sensitivities in reaction networks.ConclusionsAs a gradient-free method, the proposed sensitivity analysis provides a significant advantage when dealing with complex stochastic systems with a large number of parameters. In addition, the knowledge of the structure of the FIM can allow to efficiently address questions on parameter identifiability, estimation and robustness. The proposed method is tested and validated on three biochemical systems, namely: (a) a protein production/degradation model where explicit solutions are available, permitting a careful assessment of the method, (b) the p53 reaction network where quasi-steady stochastic oscillations of the concentrations are observed, and for which continuum approximations (e.g. mean field, stochastic Langevin, etc.) break down due to persistent oscillations between high and low populations, and (c) an Epidermal Growth Factor Receptor model which is an example of a high-dimensional stochastic reaction network with more than 200 reactions and a corresponding number of parameters.