On the Markov chain tree theorem in the Max algebra

Gursoy, Buket Benek; Kirkland, Steve; Mason, Oliver; Sergeev, Sergĕı

doi:10.13001/1081-3810.1636

Cited by 5 publications

(4 citation statements)

References 20 publications

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“…The weight of the tree containing all e edges is defined by , where is the weight of an edge starting from X i and ending at when [ 36 ]. For systems which have both forward and backward reactions, if n J is the total number of nodes indexed by {1, 2… K } the same as states, then n K is the set of nodes carrying X K , and is the set of nodes carrying given N 0 root node of the tree Ѭ, then the walk from one node to another node is given by: …”

Section: Resultsmentioning

confidence: 99%

Novel domain expansion methods to improve the computational efficiency of the Chemical Master Equation solution for large biological networks

2020

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Background Numerical 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. Results To 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. Conclusions The 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.

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Section: Resultsmentioning

confidence: 99%

Novel domain expansion methods to improve the computational efficiency of the Chemical Master Equation solution for large biological networks

2020

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“…The weight of the tree containing all edges is defined by (Ѭ) = ∏ ( ) ∈Ѭ , where ( ) = ( , ′ ) = , is the weight of an edge starting from and ending at ′ when ∈ Ѭ [35]. For the systems having both forward and backward reactions, if is the total number of nodes indexed by {1,2 … } same as states, be the set of nodes carrying , and ′ be the set of nodes carrying ′ given 0 root node of the tree Ѭ, then the walk from one node to another node is given by:…”

Section: Letmentioning

confidence: 99%

Novel domain expansion methods to improve the computational efficiency of the solution of Chemical Master Equation for large biological networks.

Kosarwal¹,

Kulasiri²,

Samarasinghe³

2020

Preprint

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Background: Numerical solutions of the chemical master equation (CME) are important to understand the stochasticity of biochemical systems. However, solving CMEs is a formidable task due to the nonlinear nature of the reactions and size of the networks that result in different realisations and, most importantly, the exponential growth of the size of the state-space with respect to the number of different species in the system. When the size of the biochemical system is very large in terms of the number of variables, the solution to the CME becomes intractable. Therefore, 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 to describe the dynamic behaviour of the system. 𝐼𝑆𝑃 is based on a state-space search and the data structure standards of artificial intelligence (𝐴𝐼) 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. Results: To show the acceptability and effectiveness of our method, we apply the 𝐼𝑆𝑃 to several biological models previously discussed in the literature. According to the results of our computational experiments, we show that 𝐼𝑆𝑃 is effective in terms of speed and accuracy of the expansion, accuracy of the solution, and provides a better understanding of the state-space of the system in terms of blueprint patterns. Conclusions: The 𝐼𝑆𝑃 is the de-novo method to address the accuracy as well as the performance problems for the solution of the CME. It systematically expands the projection space based on predefined inputs, which are useful in providing accuracy in the approximation and an exact analytical solution at the time of interest. The 𝐼𝑆𝑃 was more effective in terms of predicting the behaviour of the state-space of the system and in performance management, which is a vital step towards modelling large biochemical systems.

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“…Finally, we present the result on computing the stationary distribution of an irreducible stochastic matrix. This is a version of the Markov chain tree theorem [ 22 ], formulated using the results from matrix theory (cf. [ 23 ]).…”

Section: Necessary Ingredients From Matrix Algebramentioning

confidence: 99%

“…Assuming that π 4 = 0.25 one can compute the remaining probabilities using Equation(22): π 1 ≈ 0.97, π 2 ≈ 0.83, and π 3 = 0.5. Finally, the endemic frequencies are[v 1 , v 2 , v 3 , v4 ] = [0.675, 0.225, 0.075, 0.025].…”

mentioning

confidence: 99%

Within-Host Phenotypic Evolution and the Population-Level Control of Chronic Viral Infections by Treatment and Prophylaxis

Gromov

Romero-Severson

2020

Mathematics

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Chronic viral infections can persist for decades spanning thousands of viral generations, leading to a highly diverse population of viruses with its own complex evolutionary history. We propose an expandable mathematical framework for understanding how the emergence of genetic and phenotypic diversity affects the population-level control of those infections by both non-curative treatment and chemo-prophylactic measures. Our frameworks allows both neutral and phenotypic evolution, and we consider the specific evolution of contagiousness, resistance to therapy, and efficacy of prophylaxis. We compute both the controlled and uncontrolled, population-level basic reproduction number accounting for the within-host evolutionary process where new phenotypes emerge and are lost in infected persons, which we also extend to include both treatment and prophylactic control efforts. We used these results to discuss the conditions under which the relative efficacy of prophylactic versus therapeutic methods of control are superior. Finally, we give expressions for the endemic equilibrium of these models for certain constrained versions of the within-host evolutionary model providing a potential method for estimating within-host evolutionary parameters from population-level genetic sequence data.

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On the Markov chain tree theorem in the Max algebra

Abstract: Abstract. A max-algebraic analogue of the Markov Chain Tree Theorem is presented, and its connections with the classical Markov Chain Tree Theorem and the max-algebraic spectral theory are investigated.

Cited by 5 publications

References 20 publications

Novel domain expansion methods to improve the computational efficiency of the Chemical Master Equation solution for large biological networks

Novel domain expansion methods to improve the computational efficiency of the Chemical Master Equation solution for large biological networks

Novel domain expansion methods to improve the computational efficiency of the solution of Chemical Master Equation for large biological networks.

Within-Host Phenotypic Evolution and the Population-Level Control of Chronic Viral Infections by Treatment and Prophylaxis

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