This document presents a mathematical model using Boltzmann machines to simulate the infection process of the lambda phage in E. coli, focusing on the genetic loading and expression levels of cro and cI genes.
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[...] Formal chemical reaction networks can implement Boltzmann machines, seen as a flexible class of Markov random fields, capable of generating diverse distributions and for which conditioning on data has simple physical interpretations. Boltzmann machines are an established model of probabilistic neural networks due to their analytical tractability and connections to spin systems in statistical physics and Hopfield networks in computer science. As we have seen above, Boltzmann machines are used in a wide range of applications. - Boltzmann Machines and Their Utility in Biology Boltzmann machines are a class of binary stochastic neural networks, meaning that each node randomly switches between the values 0 and 1 according to a specified distribution. [...]
[...] est the free energy difference between the total free energy to occupy simultaneously two adjacent sites i and j and the sum of the intrinsic binding free energies and In the modeling, the Boltzmann weight associated with the state is: The Boltzmann weight associated with the state is: And so on . The probability of a given configuration n is expressed by: With : et the concentration of free dimers repressors. On defines the probabilities that the promoters PRM and PR are OFF: Figure 1 : prediction of repression curves in PR and PRM. [...]
[...] The genetic loading of the lambda phage is described by a mathematical model that takes into account the expression levels of the cro and cI genes. Due to the fact that in prokaryotes, genes are activated and deactivated (OFF) by regulatory proteins that interact with DNA sequences, a system is obtained that oscillates randomly between two phenotypes. The mathematical model models the stochastic behavior of this switch using Markov chains, following the random evolution of the number of proteins produced by the cro and cI genes, taking into account the promoter states. [...]
[...] Boltzmann Machines I. General Presentation of Boltzmann Machines In a microscale environment, molecular accounts are weak and a real (or synthetic) number will have to respond to internal and environmental signals. Probabilistic inference using Boltzmann chemical machines provides a framework for how this can be achieved. Artificial cellular systems, like unicellular organisms, must make informed decisions based on information from their environment. They need this to successfully complete complex tasks such as: finding and exploiting food sources, avoiding toxins and predators, and navigating critical stages of their life cycle. [...]
[...] Implementation of the Boltzmann machine in a molecular system We can use PyTorch which is a Python programming language that helps with the construction and execution of learning networks. We can choose to use the architecture of the restricted Boltzmann machine: [...]
