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DeepCME: A deep learning framework for computing solution statistics of the chemical master equation.

ABSTRACT: Stochastic models of biomolecular reaction networks are commonly employed in systems and synthetic biology to study the effects of stochastic fluctuations emanating from reactions involving species with low copy-numbers. For such models, the Kolmogorov's forward equation is called the chemical master equation (CME), and it is a fundamental system of linear ordinary differential equations (ODEs) that describes the evolution of the probability distribution of the random state-vector representing the copy-numbers of all the reacting species. The size of this system is given by the number of states that are accessible by the chemical system, and for most examples of interest this number is either very large or infinite. Moreover, approximations that reduce the size of the system by retaining only a finite number of important chemical states (e.g. those with non-negligible probability) result in high-dimensional ODE systems, even when the number of reacting species is small. Consequently, accurate numerical solution of the CME is very challenging, despite the linear nature of the underlying ODEs. One often resorts to estimating the solutions via computationally intensive stochastic simulations. The goal of the present paper is to develop a novel deep-learning approach for computing solution statistics of high-dimensional CMEs by reformulating the stochastic dynamics using Kolmogorov's backward equation. The proposed method leverages superior approximation properties of Deep Neural Networks (DNNs) to reliably estimate expectations under the CME solution for several user-defined functions of the state-vector. This method is algorithmically based on reinforcement learning and it only requires a moderate number of stochastic simulations (in comparison to typical simulation-based approaches) to train the "policy function". This allows not just the numerical approximation of various expectations for the CME solution but also of its sensitivities with respect to all the reaction network parameters (e.g. rate constants). We provide four examples to illustrate our methodology and provide several directions for future research.

SUBMITTER: Gupta A

PROVIDER: S-EPMC8687598 | biostudies-literature |

REPOSITORIES: biostudies-literature

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Project description:The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to "lift" this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging hp-discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the "basis" of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage savings over direct approaches.

Project description:BackgroundNumerical solutions of the chemical master equation (CME) are important for understanding the stochasticity of biochemical systems. However, solving CMEs is a formidable task. This task is complicated due to the nonlinear nature of the reactions and the size of the networks which result in different realizations. Most importantly, the exponential growth of the size of the state-space, with respect to the number of different species in the system makes this a challenging assignment. When the biochemical system has a large number of variables, the CME solution becomes intractable. We introduce the intelligent state projection (ISP) method to use in the stochastic analysis of these systems. For any biochemical reaction network, it is important to capture more than one moment: this allows one to describe the system's dynamic behaviour. ISP is based on a state-space search and the data structure standards of artificial intelligence (AI). It can be used to explore and update the states of a biochemical system. To support the expansion in ISP, we also develop a Bayesian likelihood node projection (BLNP) function to predict the likelihood of the states.ResultsTo demonstrate the acceptability and effectiveness of our method, we apply the ISP method to several biological models discussed in prior literature. The results of our computational experiments reveal that the ISP method is effective both in terms of the speed and accuracy of the expansion, and the accuracy of the solution. This method also provides a better understanding of the state-space of the system in terms of blueprint patterns.ConclusionsThe ISP is the de-novo method which addresses both accuracy and performance problems for CME solutions. It systematically expands the projection space based on predefined inputs. This ensures accuracy in the approximation and an exact analytical solution for the time of interest. The ISP was more effective both in predicting the behavior of the state-space of the system and in performance management, which is a vital step towards modeling large biochemical systems.

Project description:Fog computing (FC) based sensor networks have emerged as a propitious archetype for next-generation wireless communication technology with caching, communication, and storage capacity services in the edge. Mobile edge computing (MEC) is a new era of digital communication and has a rising demand for intelligent devices and applications. It faces performance deterioration and quality of service (QoS) degradation problems, especially in the Internet of Things (IoT) based scenarios. Therefore, existing caching strategies need to be enhanced to augment the cache hit ratio and manage the limited storage to accelerate content deliveries. Alternatively, quantum computing (QC) appears to be a prospect of more or less every typical computing problem. The framework is basically a merger of a deep learning (DL) agent deployed at the network edge with a quantum memory module (QMM). Firstly, the DL agent prioritizes caching contents via self organizing maps (SOMs) algorithm, and secondly, the prioritized contents are stored in QMM using a Two-Level Spin Quantum Phenomenon (TLSQP). After selecting the most appropriate lattice map (32 × 32) in 750,000 iterations using SOMs, the data points below the dark blue region are mapped onto the data frame to get the videos. These videos are considered a high priority for trending according to the input parameters provided in the dataset. Similarly, the light-blue color region is also mapped to get medium-prioritized content. After the SOMs algorithm's training, the topographic error (TE) value together with quantization error (QE) value (i.e., 0.0000235) plotted the most appropriate map after 750,000 iterations. In addition, the power of QC is due to the inherent quantum parallelism (QP) associated with the superposition and entanglement principles. A quantum computer taking "n" qubits that can be stored and execute 2 n presumable combinations of qubits simultaneously reduces the utilization of resources compared to conventional computing. It can be analyzed that the cache hit ratio will be improved by ranking the content, removing redundant and least important content, storing the content having high and medium prioritization using QP efficiently, and delivering precise results. The experiments for content prioritization are conducted using Google Colab, and IBM's Quantum Experience is considered to simulate the quantum phenomena.

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DeepCME: A deep learning framework for computing solution statistics of the chemical master equation.

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