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Problem
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problem background main
problem description main
Here we study a model of heteropolymers that perform template-assisted ligation based on Watson-Crick-like hybridization, which can display a reduction of information entropy along with increasing complexity. Consider a general case of heteropolymers with Z types of monomers capable of making Z/2 mutually complementary pairs (noted as i and i*) going through cycles, where during the "night" phase of each cycle, existing heteropolymers may serve as templates for ligation of pairs of chains to form longer ones, and during the "day" phase, all hybridized pairs dissociate and individual chains are fully dispersed. During the ligation process, we assume that if a "2-mer sequence" (noted as ij) appears anywhere within a polymer, then it can act as a template, help connecting 2 other polymers, one with j' at an end, and the other with i' at an end, to connect and form sequence j'i' (polymers here are DNA-like, meaning the 2 hybridized chains are directional and anti-parallel) during the ligation phase. Therefore, we can directly model the dynamics of every 2-mer's concentration, noted as . Given the initial concentration of every monomer and 2-mer in the system, as well as the ligation rate and timepoints of evaluation, we can simulate 2-mer dynamics and eventually the change of information entropy of these 2-mers.
Plain-text mathematical notation (without MathML)
Here we study a model of heteropolymers that perform template-assisted ligation based on Watson-Crick-like hybridization, which can display a reduction of information entropy along with increasing complexity. Consider a general case of heteropolymers with Z types of monomers capable of making Z/2 mutually complementary pairs (noted as i and i*) going through cycles, where during the "night" phase of each cycle, existing heteropolymers may serve as templates for ligation of pairs of chains to form longer ones, and during the "day" phase, all hybridized pairs dissociate and individual chains are fully dispersed. During the ligation process, we assume that if a "2-mer sequence" (noted as ij) appears anywhere within a polymer, then it can act as a template, help connecting 2 other polymers, one with j' at an end, and the other with i' at an end, to connect and form sequence j'i' (polymers here are DNA-like, meaning the 2 hybridized chains are directional and anti-parallel) during the ligation phase. Therefore, we can directly model the dynamics of every 2-mer's concentration, noted as d_(ij). Given the initial concentration of every monomer and 2-mer in the system, as well as the ligation rate and timepoints of evaluation, we can simulate 2-mer dynamics and eventually the change of information entropy of these 2-mers.
Original LaTeX notation
Here we study a model of heteropolymers that perform template-assisted ligation based on Watson-Crick-like hybridization, which can display a reduction of information entropy along with increasing complexity. Consider a general case of heteropolymers with Z types of monomers capable of making Z/2 mutually complementary pairs (noted as i and i*) going through cycles, where during the "night" phase of each cycle, existing heteropolymers may serve as templates for ligation of pairs of chains to form longer ones, and during the "day" phase, all hybridized pairs dissociate and individual chains are fully dispersed. During the ligation process, we assume that if a "2-mer sequence" (noted as ij) appears anywhere within a polymer, then it can act as a template, help connecting 2 other polymers, one with j' at an end, and the other with i' at an end, to connect and form sequence j'i' (polymers here are DNA-like, meaning the 2 hybridized chains are directional and anti-parallel) during the ligation phase. Therefore, we can directly model the dynamics of every 2-mer's concentration, noted as $d_{ij}$. Given the initial concentration of every monomer and 2-mer in the system, as well as the ligation rate and timepoints of evaluation, we can simulate 2-mer dynamics and eventually the change of information entropy of these 2-mers.problem io
''' Inputs: c0: concentration of monomers. numpy array with dimensions [Z] d0: concentration of 2-mers, numpy array with dimensions [Z, Z] lig: ligation rate of any combination of monomers when forming a 2-mer, numpy array with dimensions [Z, Z] tf: timespan of simulation, float nsteps: number of steps of simulation, int Outputs: entropy_list: information entropy of 2-mers at each timestep, numpy array of length nsteps '''
problem name
two_mer_entropy
required dependencies
import numpy as np from math import exp from scipy.integrate import solve_ivp
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initial import