Kepa Ruiz-Mirazo,†,∥ Carlos Briones,‡,∥ and Andreś de la Escosura* †Biophysics Unit (CSIC-UPV/EHU), Leioa, and Department of Logic and Philosophy of Science, University of the Basque Country, Avenida de Tolosa 70, 20080 Donostia−San Sebastiań, Spain ‡Department of Molecular Evolution, Centro de Astrobiología (CSIC−INTA, associated to the NASA Astrobiology Institute), Carretera de Ajalvir, Km 4, 28850 Torrejoń de Ardoz, Madrid, Spain Organic Chemistry Department, Universidad Autońoma de Madrid, Cantoblanco, 28049 Madrid, Spain

Prebiotic systems chemistry: new perspectives for the origins of life.

Stochastic fluctuations of molecule numbers are ubiquitous in biological systems. Important examples include gene expression and enzymatic processes in living cells. Such systems are typically modelled as chemical reaction networks whose dynamics are governed by the Chemical Master Equation. Despite its simple structure, no analytic solutions to the Chemical Master Equation are known for most systems. Moreover, stochastic simulations are computationally expensive, making systematic analysis and statistical inference a challenging task. Consequently, significant effort has been spent in recent decades on the development of efficient approximation and inference methods. This article gives an introduction to basic modelling concepts as well as an overview of state of the art methods. First, we motivate and introduce deterministic and stochastic methods for modelling chemical networks, and give an overview of simulation and exact solution methods. Next, we discuss several approximation methods, including the chemical Langevin equation, the system size expansion, moment closure approximations, time-scale separation approximations and hybrid methods. We discuss their various properties and review recent advances and remaining challenges for these methods. We present a comparison of several of these methods by means of a numerical case study and highlight some of their respective advantages and disadvantages. Finally, we discuss the problem of inference from experimental data in the Bayesian framework and review recent methods developed the literature. In summary, this review gives a self-contained introduction to modelling, approximations and inference methods for stochastic chemical kinetics.

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Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

Stochastic mechano-chemical kinetics of molecular motors: A multidisciplinary enterprise from a physicist’s perspective

Approximation and inference methods for stochastic biochemical kinetics - a tutorial review

Background: It is well known that the deterministic dynamics of biochemical reaction networks can be more easily studied if timescale separation conditions are invoked (the quasi-steady-state assumption). In this case the deterministic dynamics of a large network of elementary reactions are well described by the dynamics of a smaller network of effective reactions. Each of the latter represents a group of elementary reactions in the large network and has associated with it an effective macroscopic rate law. A popular method to achieve model reduction in the presence of intrinsic noise consists of using the effective macroscopic rate laws to heuristically deduce effective probabilities for the effective reactions which then enables simulation via the stochastic simulation algorithm (SSA). The validity of this heuristic SSA method is a priori doubtful because the reaction probabilities for the SSA have only been rigorously derived from microscopic physics arguments for elementary reactions. Results: We here obtain, by rigorous means and in closed-form, a reduced linear Langevin equation description of the stochastic dynamics of monostable biochemical networks in conditions characterized by small intrinsic noise and timescale separation. The slow-scale linear noise approximation (ssLNA), as the new method is called, is used to calculate the intrinsic noise statistics of enzyme and gene networks. The results agree very well with SSA simulations of the non-reduced network of elementary reactions. In contrast the conventional heuristic SSA is shown to overestimate the size of noise for Michaelis-Menten kinetics, considerably under-estimate the size of noise for Hill-type kinetics and in some cases even miss the prediction of noise-induced oscillations. Conclusions: A new general method, the ssLNA, is derived and shown to correctly describe the statistics of intrinsic noise about the macroscopic concentrations under timescale separation conditions. The ssLNA provides a simple and accurate means of performing stochastic model reduction and hence it is expected to be of widespread utility in studying the dynamics of large noisy reaction networks, as is common in computational and systems biology.

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The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions

We describe a mechanism for pronounced biochemical oscillations, relevant to microscopic systems, such as the intracellular environment. This mechanism operates for reaction schemes which, when modeled using deterministic rate equations, fail to exhibit oscillations for any values of rate constants. The mechanism relies on amplification of the underlying stochasticity of reaction kinetics within a narrow window of frequencies. This amplification means that fluctuations have a dominant effect, even though the number of molecules in the system is relatively large. The mechanism is quantitatively studied within simple models of self-regulatory gene expression, and glycolytic oscillations.

https://www.theory.physics.manchester.ac.uk/~ajm/mnns07.pdf

Amplified Biochemical Oscillations in Cellular Systems

The ability to dynamically probe single enzymes allows the experimental investigation of enzyme kinetics with unprecedented resolution. In this paper we develop a simple theory which predicts that certain classes of enzyme pathways can be distinguished by studying the turnover rate, V, as a function of substrate concentration, [S]. In particular, we study the steady state of a single enzyme interacting with a bath of substrate molecules, and analyse it as a stochastic process. The V([S]) relation is found to depend sensitively on the manner in which substrate molecules in the bath are replenished. We focus on a gedanken experiment in which the average substrate concentration is kept fixed by allowing molecules to enter the bath at a constant rate. We derive the exact relationship between V and [S], which has a relatively simple form, though different to that of the Michaelis–Menten (MM) equation. Interestingly, the MM equation is exactly recovered if the substrate concentration is instantaneously maintained with molecular precision. We examine the new V([S]) relation for a number of enzyme pathways and find that it differentiates between enzyme reactions involving one or many intermediate enzyme–substrate complexes. This, in principle, allows one to probe the internal conformations of enzymes by careful measurement of V([S]) curves in appropriately designed single enzyme experiments.

https://www.theory.physics.manchester.ac.uk/~ajm/smn05.pdf

Single enzyme pathways and substrate fluctuations

http://absimage.aps.org/image/MAR05/MWS_MAR05-2004-001584.pdf

M. O. Stefanini

Papers

Amplified Biochemical Oscillations in Cellular Systems

Single enzyme pathways and substrate fluctuations

Single Enzyme Pathways and Substrate Fluctuations