scholarly journals Quantum Techniques for Reaction Networks

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
John C. Baez
Keyword(s):  
Master Equation ◽  
Rate Equation ◽  
Quantum Physics ◽  
Time Derivative ◽  
Reaction Network ◽  
Reaction Networks ◽  
Expected Number ◽  
Second Quantization ◽  
Stochastic Time ◽  
Number Of Particles

Reaction networks are a general formalism for describing collections of classical entities interacting in a random way. While reaction networks are mainly studied by chemists, they are equivalent to Petri nets, which are used for similar purposes in computer science and biology. As noted by Doi and others, techniques from quantum physics, such as second quantization, can be adapted to apply to such systems. Here we use these techniques to study how the “master equation” describing stochastic time evolution for a reaction network is related to the “rate equation” describing the deterministic evolution of the expected number of particles of each species in the large-number limit. We show that the relation is especially strong when a solution of master equation is a “coherent state”, meaning that the numbers of entities of each kind are described by independent Poisson distributions. Remarkably, in this case the rate equation and master equation give the exact same formula for the time derivative of the expected number of particles of each species.

scholarly journals Uncertainty quantification for quantum chemical models of complex reaction networks

2016 ◽  
Vol 195 ◽  
pp. 497-520 ◽  
Author(s):  
Jonny Proppe ◽  
Tamara Husch ◽  
Gregor N. Simm ◽  
Markus Reiher
Keyword(s):  
Free Energy ◽  
Electronic Structure ◽  
State Space ◽  
Time Scales ◽  
Reaction Network ◽  
Reaction Networks ◽  
Multiple Time Scales ◽  
Complex Reaction ◽  
Discrete State ◽  
Continuous State

For the quantitative understanding of complex chemical reaction mechanisms, it is, in general, necessary to accurately determine the corresponding free energy surface and to solve the resulting continuous-time reaction rate equations for a continuous state space. For a general (complex) reaction network, it is computationally hard to fulfill these two requirements. However, it is possible to approximately address these challenges in a physically consistent way. On the one hand, it may be sufficient to consider approximate free energies if a reliable uncertainty measure can be provided. On the other hand, a highly resolved time evolution may not be necessary to still determine quantitative fluxes in a reaction network if one is interested in specific time scales. In this paper, we present discrete-time kinetic simulations in discrete state space taking free energy uncertainties into account. The method builds upon thermo-chemical data obtained from electronic structure calculations in a condensed-phase model. Our kinetic approach supports the analysis of general reaction networks spanning multiple time scales, which is here demonstrated for the example of the formose reaction. An important application of our approach is the detection of regions in a reaction network which require further investigation, given the uncertainties introduced by both approximate electronic structure methods and kinetic models. Such cases can then be studied in greater detail with more sophisticated first-principles calculations and kinetic simulations.

scholarly journals Improved estimations of stochastic chemical kinetics by finite-state expansion

2021 ◽  
Vol 477 (2251) ◽  
pp. 20200964
Author(s):  
Tabea Waizmann ◽  
Luca Bortolussi ◽  
Andrea Vandin ◽  
Mirco Tribastone
Keyword(s):  
Master Equation ◽  
State Space ◽  
Stochastic Dynamics ◽  
Reaction Network ◽  
Discrete State ◽  
State Expansion ◽  
Analytical Approximations ◽  
Finite State ◽  
Stochastic Reaction ◽  
Discrete State Space

Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite-state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations and multi-stability.

Feedback mechanisms in mineral replacement networks: an experimental investigation of the ultramafic model system

2021 ◽  
Author(s):  
Ingvild Aarrestad ◽  
Oliver Plümper ◽  
Desiree Roerdink ◽  
Andreas Beinlich
Keyword(s):  
Reaction Rate ◽  
Energy Transport ◽  
Kinetic Modelling ◽  
Indirect Evidence ◽  
Reaction Network ◽  
Reaction Networks ◽  
Silicate System ◽  
Rock Interaction ◽  
Feedback Mechanisms ◽  
Salinity Effect

<p>The overall rates of multi-component reaction networks are known to be controlled by feedback mechanisms. Feedback mechanisms represent loop systems where the output of the system is conveyed back as input and the system is either accelerated or regulated (positive and negative feedback respectively). In other words, feedback mechanisms control the rate of a reaction network without external influences. Feedback mechanisms are well-studied in a variety of reaction networks (e.g. bio-chemical, atmospheric); however, in fluid-rock interaction systems they are not researched as such. Still, indirect evidence, theoretical considerations and direct observations attest to their existence [e.g. 1, 2, 3]. It remains unknown how mass and energy transport between distinct reaction sites affect the overall reaction rate and outcome through feedback mechanisms. We propose that feedback mechanisms are a missing critical ingredient to understand reaction progress and timescales of fluid-rock interactions. We apply the serpentinization of ultramafic silicates as a relatively simple reaction network to investigate feedback mechanisms during fluid-rock interactions. Recent studies show that theoretical timescale-predictions appear inconsistent with natural observations [e.g. 4, 5]. The ultramafic silicate system is ideal for investigating feedback mechanisms as it is relevant to natural processes, is reactive on timescales that can be explored in the laboratory, and natural peridotite typically consists of less than four phases. Our preliminary observations indicate a feedback between pyroxene dissolution and olivine serpentinization. Olivine serpentinization appears to proceed faster in the presence of pyroxene. Furthermore, the bulk system reaction rate increases with increasing fluid salinity, which is opposite to the salinity effect on the monomineralic olivine system. Dunite (>90% olivine) is rare, which is why it is crucial to explore the more common pyroxene-bearing systems. The salinity effect is important to investigate due to the inevitable increase in fluid salinity from the boiling-induced phase separation and OH-uptake in the formation of serpentine. Here we present preliminary textural and chemical observations, which will subsequently be used for kinetic modelling of feedback.</p><p>[1] Ortoleva P., Merino, E., Moore, C. & Chadam, J. (1987). American Journal of Science <strong>287</strong>, 997-1007.</p><p>[2] Centrella, S., Austrheim, H., & Putnis, A. (2015). Lithos <strong>236–237</strong>, 245–255.</p><p>[3] Nakatani, T. & Nakamura, M. (2016). Geochemistry, Geophysics, Geosystems <strong>17</strong>, 3393-3419.</p><p>[4] Ingebritsen, S. E. & Manning, C. E. (2010). Geofluids <strong>10</strong>, 193-205.</p><p>[5] Beinlich, A., John, T., Vrijmoed, J.C., Tominaga, M., Magna, T. & Podladchikov, Y.Y. (2020). Nature Geoscience <strong>13</strong>, 307–311.</p>

scholarly journals A compositional framework for reaction networks

2017 ◽  
Vol 29 (09) ◽  
pp. 1750028 ◽  
Author(s):  
John C. Baez ◽  
Blake S. Pollard
Keyword(s):  
Dynamical System ◽  
Steady State ◽  
General Framework ◽  
Nonlinear Dynamical System ◽  
Reaction Network ◽  
Reaction Networks ◽  
Nonlinear Dynamical ◽  
Input And Output ◽  
Open Dynamical System ◽  
Steady State Solutions

Reaction networks, or equivalently Petri nets, are a general framework for describing processes in which entities of various kinds interact and turn into other entities. In chemistry, where the reactions are assigned ‘rate constants’, any reaction network gives rise to a nonlinear dynamical system called its ‘rate equation’. Here we generalize these ideas to ‘open’ reaction networks, which allow entities to flow in and out at certain designated inputs and outputs. We treat open reaction networks as morphisms in a category. Composing two such morphisms connects the outputs of the first to the inputs of the second. We construct a functor sending any open reaction network to its corresponding ‘open dynamical system’. This provides a compositional framework for studying the dynamics of reaction networks. We then turn to statics: that is, steady state solutions of open dynamical systems. We construct a ‘black-boxing’ functor that sends any open dynamical system to the relation that it imposes between input and output variables in steady states. This extends our earlier work on black-boxing for Markov processes.

scholarly journals A Chemically Consistent Graph Architecture for Massive Reaction Networks Applied to Solid-Electrolyte Interphase Formation

2020 ◽  
Author(s):  
Samuel Blau ◽  
Hetal Patel ◽  
Evan Spotte-Smith ◽  
Xiaowei Xie ◽  
Shyam Dwaraknath ◽  
...  
Keyword(s):  
Solid Electrolyte ◽  
First Principles ◽  
Low Cost ◽  
Solid Electrolyte Interphase ◽  
Negative Electrode ◽  
Reaction Network ◽  
Reaction Networks ◽  
Graph Representations ◽  
Li Ion ◽  
Shortest Path Algorithms

Modeling reactivity with chemical reaction networks could yield fundamental mechanistic understanding that would expedite the development of processes and technologies for energy storage, medicine, catalysis, and more. Thus far, reaction networks have been limited in size by chemically inconsistent graph representations of multi-reactant reactions (e.g. A + B reacts to C) that cannot enforce stoichiometric constraints, precluding the use of optimized shortest-path algorithms. Here, we report a chemically consistent graph architecture that overcomes these limitations using a novel multi-reactant representation and iterative cost-solving procedure. Our approach enables the identification of all low-cost pathways to desired products in massive reaction networks containing reactions of any stoichiometry, allowing for the investigation of vastly more complex systems than previously possible. Leveraging our architecture, we construct the first ever electrochemical reaction network from first-principles thermodynamic calculations to describe the formation of the Li-ion solid electrolyte interphase (SEI), which is critical for passivation of the negative electrode. Using this network comprised of nearly 6,000 species and 4.5 million reactions, we interrogate the formation of a key SEI component, lithium ethylene dicarbonate. We automatically identify previously proposed mechanisms as well as multiple novel pathways containing counter-intuitive reactions that have not, to our knowledge, been reported in the literature. We envision that our framework and data-driven methodology will facilitate efforts to engineer the composition-related properties of the SEI - or of any complex chemical process - through selective control of reactivity.

scholarly journals A general technique for the detection of switch-like bistability in chemical reaction networks governed by mass action kinetics with conservation laws

2020 ◽  
Author(s):  
Brandon C Reyes ◽  
Irene Otero-Muras ◽  
Vladislav A Petyuk
Keyword(s):  
Conservation Laws ◽  
Signaling Pathways ◽  
Chemical Reaction ◽  
Reaction Network ◽  
Mass Action ◽  
Numerical Continuation ◽  
Reaction Networks ◽  
General Technique ◽  
Bistable Switch ◽  
Mass Action Kinetics

AbstractBackgroundTheoretical analysis of signaling pathways can provide a substantial amount of insight into their function. One particular area of research considers signaling pathways capable of assuming two or more stable states given the same amount of signaling ligand. This phenomenon of bistability can give rise to switch-like behavior, a mechanism that governs cellular decision making. Investigation of whether or not a signaling pathway can confer bistability and switch-like behavior, without knowledge of specific kinetic rate constant values, is a mathematically challenging problem. Recently a technique based on optimization has been introduced, which is capable of finding example parameter values that confer switch-like behavior for a given pathway. Although this approach has made it possible to analyze moderately sized pathways, it is limited to reaction networks that presume a uniterminal structure. It is this limited structure we address by developing a general technique that applies to any mass action reaction network with conservation laws.ResultsIn this paper we developed a generalized method for detecting switch-like bistable behavior in any mass action reaction network with conservation laws. The method involves 1) construction of a constrained optimization problem using the determinant of the Jacobian of the underlying rate equations, 2) minimization of the objective function to search for conditions resulting in a zero eigenvalue 3) computation of a confidence level that describes if the global minimum has been found and 4) evaluation of optimization values, using either numerical continuation or directly simulating the ODE system, to verify that a bistability region exists. The generalized method has been tested on three motifs known to be capable of bistability.ConclusionsWe have developed a variation of an optimization-based method for discovery of bistability, which is not limited to the structure of the chemical reaction network. Successful completion of the method provides an S-shaped bifurcation diagram, which indicates that the network acts as a bistable switch for the given optimization parameters.

scholarly journals Stationary Equations for Non-Markovian Biochemical Systems

2018 ◽  
Author(s):  
Jiajun Zhang ◽  
Tianshou Zhou
Keyword(s):  
Master Equation ◽  
Stochastic Analysis ◽  
Waiting Times ◽  
Reaction Network ◽  
State Variables ◽  
New Approach ◽  
Reaction Systems ◽  
Linear Noise Approximation ◽  
Biochemical Systems ◽  
Markovian System

AbstractWe develop a new approach for stochastic analysis of biochemical reaction systems with arbitrary distributions of waiting times between reaction events. Specifically, we derive a stationary generalized chemical master equation for a non-Markovian reaction network. Importantly, this equation allows to transform the original non-Markovian problem into a Markovian one by introducing a mean reaction propensity function for every reaction in the network. Furthermore, we derive a stationary generalized linear noise approximation for the non-Markovian system, which is convenient to the direct estimation of the stationary noise in state variables. These derived equations can have broad applications, and exemplars of two representative non-Markovian models provide evidence of their applicability.

scholarly journals On reaction network implementations of neural networks

2021 ◽  
Vol 18 (177) ◽  
Author(s):  
David F. Anderson ◽  
Badal Joshi ◽  
Abhishek Deshpande
Keyword(s):  
Neural Networks ◽  
Fixed Points ◽  
Reaction Network ◽  
Activation Function ◽  
Reaction Networks ◽  
Mathematical Framework ◽  
Feed Forward Neural Networks ◽  
Systems Of Odes ◽  
Do So

This paper is concerned with the utilization of deterministically modelled chemical reaction networks for the implementation of (feed-forward) neural networks. We develop a general mathematical framework and prove that the ordinary differential equations (ODEs) associated with certain reaction network implementations of neural networks have desirable properties including (i) existence of unique positive fixed points that are smooth in the parameters of the model (necessary for gradient descent) and (ii) fast convergence to the fixed point regardless of initial condition (necessary for efficient implementation). We do so by first making a connection between neural networks and fixed points for systems of ODEs, and then by constructing reaction networks with the correct associated set of ODEs. We demonstrate the theory by constructing a reaction network that implements a neural network with a smoothed ReLU activation function, though we also demonstrate how to generalize the construction to allow for other activation functions (each with the desirable properties listed previously). As there are multiple types of ‘networks’ used in this paper, we also give a careful introduction to both reaction networks and neural networks, in order to disambiguate the overlapping vocabulary in the two settings and to clearly highlight the role of each network’s properties.

scholarly journals Early warning signals in chemical reaction networks

2020 ◽  
Vol 56 (26) ◽  
pp. 3725-3728
Author(s):  
Oliver R. Maguire ◽  
Albert S. Y. Wong ◽  
Jan Harm Westerdiep ◽  
Wilhelm T. S. Huck
Keyword(s):  
Complex Systems ◽  
Chemical Reaction ◽  
Early Warning ◽  
Reaction Network ◽  
Chemical Reaction Networks ◽  
Chemical Reaction Network ◽  
Reaction Networks ◽  
Warning Signals ◽  
Early Warning Signals ◽  
Complex Chemical

Many natural and man-made complex systems display early warning signals when close to an abrupt shift in behaviour. Here we show that such early warning signals appear in a complex chemical reaction network.

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