sjhaque14 / steinberg-signature

Computational tools to calculate higher-order autocorrelation functions from general graphs

Overview

This repository contains the code used to perform quantiative analysis in the upcoming manuscript "Anomalous behaviour of the Steinberg signature for detecting departure from thermodynamic equilibrium".

The Steinberg signature for detecting departure from thermodynamic equilibrium was introduced by I. Z. Steinberg in the 1986 Biophysical Journal paper "On the time reversal of noise signals". It exploits specialized "higher-order" autocorrelation functions to detect time-reversal asymmetry in stochastic signals emitted by an underlying continuous-time Markov process. These higher-order autocorrelation functions take the following form:

$$\mathcal{A}^{\alpha,\beta}(\tau) = \lim_{T \rightarrow 0} \frac{1}{T - \tau}\int_{0}^{T - \tau} f^{\alpha}(t)f^{\beta}(t + \tau) dt.$$

This repository contains code to calculate $\mathcal{A}^{\alpha,\beta}(\tau)$ and its time-reverse $\mathcal{A}^{\beta,\alpha}(\tau)$ for any continuous time, finite space, time-homogeneous Markov process. We represent Markov processes using finite, directed, labeled graphs $G$, as defined by the linear framework, a graph-theoretic approach to modeling biochemical networks at steady-state. The code included in this repository enables the user to accomplish the following:

• Calculate $\mathcal{A}^{\alpha,\beta}(\tau)$ and its time-reverse $\mathcal{A}^{\beta,\alpha}(\tau)$ for any continuous time, finite space, time-homogeneous Markov process, represented by a linear framework graph $G$.
• Calculate $\mathcal{I}^{\alpha,\beta}(G) = \int_{0}^{\infty} \left(\mathcal{A}^{\alpha,\beta}(\tau) - \mathcal{A}^{\beta,\alpha}(\tau)\right) d\tau$ as a function of the cycle affinity of the 3-vertex graph $\tilde{A}(C) = \ln{\left ( \dfrac{ade}{bfc} \right )}$ (see edge label assignments in the figure below) as a single edge label is driven further and further from its equilibrium value (i.e. for which $\tilde{A}(C) = 0$).

Prerequisites

The following Python3 libraries are used in this library. All of the below can be installed using pip or conda.

• Numpy

• Scipy.linalg

• Matplotlib

• Tqdm (optional)

Installation

The relevant functions for calculating the Steinberg signature and generating force-area curves are included in the file steinberg_utils_3vertex.py.

import numpy as np
import scipy.linalg
import matplotlib.pyplot as plt
from tqdm import tqdm

from steinberg_utils_3vertex import *

For an example of how to use the functions in steinberg_utils_3vertex.py, see the Jupyter notebook steinberg_3vertex_official.ipynb. You may also clone this repository to obtain the relevant files.

git clone https://github.com/sjhaque14/steinberg-signature

Contact

For more details, please contact Sabina J Haque at [email protected].