# Information Thermodynamics for Time Series of Signal-Response Models

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## Abstract

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## 1. Introduction

## 2. Introduction to Continuous Information Thermodynamics

#### 2.1. Entropy Production in Heat Baths

#### 2.2. Feedbacks and Information

#### 2.3. Irreversible Entropy Production

## 3. Bivariate Time Series Information Thermodynamics

#### 3.1. Setting and Definition of Causal Representations

#### 3.2. Definition of Mapping Irreversibility and the Standard Integral Fluctuation Theorem

#### 3.3. Ito Inequality for Time Series

#### 3.4. The Mapping Irreversibility Density

## 4. The Fluctuation Theorem for Time Series of Signal-Response Models

## 5. Applications

#### 5.1. The Basic Linear Response Model

#### 5.2. Receptor-Ligand Systems

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

OU | Ornstein-Uhlenbeck process |

BLRM | basic linear signal-response model |

## Appendix A. Mapping Irreversibility in the BLRM

**Figure A1.**Numerical verification of the analytical solution for the entropy production ${\Phi}_{\tau}^{xy}$ with observational time $\tau $ in the BLRM. The parameters are $\beta =0.2$ and ${t}_{rel}=10$. The slight down-deviation for small $\tau $ is due to the finite box length in the discretized space, while the up-deviation for $\tau \to \infty $ is due to the finite number of samples.

## Appendix B. Backward Transfer Entropy in the BLRM

## Appendix C. The Causal Influence Rate Converges to the Horowitz-Esposito Information Flow in the BLRM

**Figure A2.**Probability currents $\overrightarrow{J}$ in the BLRM, estimated with $\tau =0.1$. The parameters are $\beta =0.2$ and ${t}_{rel}=10$. The space coordinates are in units of the standard deviation.

## Appendix D. Numerical Convergence of the Mapping Irreversibility to the Entropy Production in the Feedback Cooling Model

## Appendix E. Numerical Estimation of the Entropy Production in the Bivariate Gaussian Approximation

## References

- Jarzynski, C. Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale. Annu. Rev. Condens. Matter Phys.
**2011**, 2, 329–351. [Google Scholar] [CrossRef] - Parrondo, J.M.; van den Broeck, C.; Kawai, R. Entropy production and the arrow of time. New J. Phys.
**2009**, 11, 073008. [Google Scholar] [CrossRef][Green Version] - Feng, E.H.; Crooks, G.E. Length of time’s arrow. Phys. Revi. Lett.
**2008**, 101, 090602. [Google Scholar] [CrossRef] [PubMed] - Jarzynski, C. Nonequilibrium equality for free energy differences. Phys. Rev. Lett.
**1997**, 78, 2690. [Google Scholar] [CrossRef] - Crooks, G.E. Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E
**1999**, 60, 2721. [Google Scholar] [CrossRef] - Evans, D.J.; Searles, D.J. The fluctuation theorem. Adv. Phys.
**2002**, 51, 1529–1585. [Google Scholar] [CrossRef] - Kawai, R.; Parrondo, J.; van den Broeck, C. Dissipation: The phase-space perspective. Phys. Rev. Lett.
**2007**, 98, 080602. [Google Scholar] [CrossRef] [PubMed] - Jarzynski, C. Hamiltonian derivation of a detailed fluctuation theorem. J. Stat. Phys.
**2000**, 98, 77–102. [Google Scholar] [CrossRef] - Chernyak, V.Y.; Chertkov, M.; Jarzynski, C. Path-integral analysis of fluctuation theorems for general Langevin processes. J. Stat. Mech. Theory Exp.
**2006**, 2006, P08001. [Google Scholar] [CrossRef] - Ito, S.; Sagawa, T. Information thermodynamics on causal networks. Phys. Rev. Lett.
**2013**, 111, 180603. [Google Scholar] [CrossRef] - Sagawa, T.; Ueda, M. Nonequilibrium thermodynamics of feedback control. Phys. Rev. E
**2012**, 85, 021104. [Google Scholar] [CrossRef] - Sagawa, T.; Ueda, M. Generalized Jarzynski equality under nonequilibrium feedback control. Phys. Rev. Lett.
**2010**, 104, 090602. [Google Scholar] [CrossRef] - Szilard, L. On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings. Syst. Res. Behav. Sci.
**1964**, 9, 301–310. [Google Scholar] [CrossRef] - Martínez, I.A.; Roldán, É.; Dinis, L.; Petrov, D.; Parrondo, J.M.; Rica, R.A. Brownian carnot engine. Nat. Phys.
**2016**, 12, 67–70. [Google Scholar] [CrossRef] [PubMed] - Ciliberto, S. Experiments in stochastic thermodynamics: Short history and perspectives. Phys. Rev. X
**2017**, 7, 021051. [Google Scholar] [CrossRef] - Toyabe, S.; Sagawa, T.; Ueda, M.; Muneyuki, E.; Sano, M. Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. Nat. Phys.
**2010**, 6, 988–992. [Google Scholar] [CrossRef][Green Version] - Koski, J.V.; Maisi, V.F.; Pekola, J.P.; Averin, D.V. Experimental realization of a Szilard engine with a single electron. Proc. Natl. Acad. Sci. USA
**2014**, 111, 13786–13789. [Google Scholar] [CrossRef] [PubMed] - Horowitz, J.M.; Esposito, M. Thermodynamics with continuous information flow. Phys. Rev. X
**2014**, 4, 031015. [Google Scholar] [CrossRef] - Rosinberg, M.L.; Horowitz, J.M. Continuous information flow fluctuations. EPL (Europhysics Letters)
**2016**, 116, 10007. [Google Scholar] [CrossRef][Green Version] - Horowitz, J.M. Multipartite information flow for multiple Maxwell demons. J. Stat. Mech. Theory Exp.
**2015**, 2015, P03006. [Google Scholar] [CrossRef] - Ito, S. Backward transfer entropy: Informational measure for detecting hidden Markov models and its interpretations in thermodynamics, gambling and causality. Sci. Rep.
**2016**, 6. [Google Scholar] [CrossRef] [PubMed] - Spinney, R.E.; Lizier, J.T.; Prokopenko, M. Transfer entropy in physical systems and the arrow of time. Phys. Rev. E
**2016**, 94, 022135. [Google Scholar] [CrossRef] [PubMed] - Parrondo, J.M.; Horowitz, J.M.; Sagawa, T. Thermodynamics of information. Nat. Phys.
**2015**, 11, 131–139. [Google Scholar] [CrossRef] - Ito, S. Unified framework for the second law of thermodynamics and information thermodynamics based on information geometry. arXiv, 2018; arXiv:1810.09545. [Google Scholar]
- Ito, S. Stochastic thermodynamic interpretation of information geometry. Phys. Rev. Lett.
**2018**, 121, 030605. [Google Scholar] [CrossRef] [PubMed] - Roldán, É.; Parrondo, J.M. Entropy production and Kullback-Leibler divergence between stationary trajectories of discrete systems. Phys. Rev. E
**2012**, 85, 031129. [Google Scholar] [CrossRef] [PubMed] - Cover, T.M.; Thomas, J.A. Elements of Information Theory; John Wiley & Sons: Hoboken, NJ, USA, 2012. [Google Scholar]
- Sekimoto, K. Langevin equation and thermodynamics. Prog. Theor. Phys. Supp.
**1998**, 130, 17–27. [Google Scholar] [CrossRef] - Crooks, G.E.; Still, S.E. Marginal and conditional second laws of thermodynamics. arXiv, 2016; arXiv:1611.04628. [Google Scholar]
- Auconi, A.; Giansanti, A.; Klipp, E. Causal influence in linear Langevin networks without feedback. Phys. Rev. E
**2017**, 95, 042315. [Google Scholar] [CrossRef] - Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys.
**2012**, 75, 126001. [Google Scholar] [CrossRef][Green Version] - Shreve, S.E. Stochastic Calculus for Finance II: Continuous-Time Models; Springer Science & Business Media: New York, NY, USA, 2004; Volume 11. [Google Scholar]
- Taniguchi, T.; Cohen, E. Onsager-Machlup theory for nonequilibrium steady states and fluctuation theorems. J. Stat. Phys.
**2007**, 126, 1–41. [Google Scholar] [CrossRef] - Seifert, U. Entropy production along a stochastic trajectory and an integral fluctuation theorem. Phys. Rev. Lett.
**2005**, 95, 040602. [Google Scholar] [CrossRef] - Risken, H. Fokker-planck equation. In The Fokker-Planck Equation; Springer: Berlin, Germany, 1996; pp. 63–95. [Google Scholar]
- Horowitz, J.M.; Sandberg, H. Second-law-like inequalities with information and their interpretations. New J. Phys.
**2014**, 16, 125007. [Google Scholar] [CrossRef][Green Version] - Gomez-Marin, A.; Parrondo, J.M.; van den Broeck, C. Lower bounds on dissipation upon coarse graining. Phys. Rev. E
**2008**, 78, 011107. [Google Scholar] [CrossRef] [PubMed] - Ito, S.; Sagawa, T. Information flow and entropy production on Bayesian networks. In Mathematical Foundations and Applications of Graph Entropy; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2016; pp. 63–100. [Google Scholar]
- Uhlenbeck, G.E.; Ornstein, L.S. On the theory of the Brownian motion. Phys. Rev.
**1930**, 36, 823. [Google Scholar] [CrossRef] - Gillespie, D.T. Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral. Phys. Rev. E
**1996**, 54, 2084. [Google Scholar] [CrossRef] - R Core Team. R: A Language and Environment for Statistical Computing; Foundation for Statistical Computing: Vienna, Austria, 2014; Available online: www.r-project.org (accessed on 14 February 2019).
- Barrett, A.B. Exploration of synergistic and redundant information sharing in static and dynamical Gaussian systems. Phys. Rev. E
**2015**, 91, 052802. [Google Scholar] [CrossRef] [PubMed] - Klipp, E.; Liebermeister, W.; Wierling, C.; Kowald, A.; Herwig, R. Systems Biology: A Textbook; John Wiley & Sons; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2016. [Google Scholar]
- Kholodenko, B.N. Cell-signalling dynamics in time and space. Nat. Rev. Mol. Cell Biol.
**2006**, 7, 165. [Google Scholar] [CrossRef] [PubMed] - Bialek, W.; Setayeshgar, S. Physical limits to biochemical signaling. Proc. Natl. Acad. Sci. USA
**2005**, 102, 10040–10045. [Google Scholar] [CrossRef][Green Version] - Tkačik, G.; Callan, C.G.; Bialek, W. Information flow and optimization in transcriptional regulation. Proc. Natl. Acad. Sci. USA
**2008**, 105, 12265–12270. [Google Scholar] [CrossRef][Green Version] - Crisanti, A.; de Martino, A.; Fiorentino, J. Statistics of optimal information flow in ensembles of regulatory motifs. Phys. Rev. E
**2018**, 97, 022407. [Google Scholar] [CrossRef][Green Version] - Waltermann, C.; Klipp, E. Information theory based approaches to cellular signaling. Biochimica et Biophysica Acta (BBA)-General Subjects
**2011**, 1810, 924–932. [Google Scholar] [CrossRef] - Di Talia, S.; Wieschaus, E.F. Short-term integration of Cdc25 dynamics controls mitotic entry during Drosophila gastrulation. Dev. Cell
**2012**, 22, 763–774. [Google Scholar] [CrossRef] [PubMed] - Nemenman, I. Gain control in molecular information processing: Lessons from neuroscience. Phys. Biol.
**2012**, 9, 026003. [Google Scholar] [CrossRef] [PubMed] - Di Talia, S.; Wieschaus, E.F. Simple biochemical pathways far from steady state can provide switchlike and integrated responses. Biophys. J.
**2014**, 107, L1–L4. [Google Scholar] [CrossRef] [PubMed] - Tkačik, G.; Walczak, A.M.; Bialek, W. Optimizing information flow in small genetic networks. Phys. Rev. E
**2009**, 80, 031920. [Google Scholar] [CrossRef] [PubMed] - Jarzynski, C. Rare events and the convergence of exponentially averaged work values. Phys. Rev. E
**2006**, 73, 046105. [Google Scholar] [CrossRef] [PubMed] - Ito, S.; Sagawa, T. Maxwell’s demon in biochemical signal transduction with feedback loop. Nat. Commun.
**2015**, 6, 7498. [Google Scholar] [CrossRef] [PubMed][Green Version] - Barnett, L.; Barrett, A.B.; Seth, A.K. Granger causality and transfer entropy are equivalent for Gaussian variables. Phys. Rev. Lett.
**2009**, 103, 238701. [Google Scholar] [CrossRef] - Duchi, J. Derivations for Linear Algebra and Optimization. Available online: https://docplayer.net/30887339-Derivations-for-linear-algebra-and-optimization.html (accessed on 11 February 2019).

**Figure 1.**Complete causal representation. The arrows represent the way we factorize the joint probability density. In the complete case the causal representation is fully degenerate: $p\left({\zeta}_{\tau}^{xy}\right)=p({x}_{t},{y}_{t})\xb7p({x}_{t+\tau},{y}_{t+\tau}|{x}_{t},{y}_{t})=p({x}_{t},{y}_{t})\xb7p\left({x}_{t+\tau}\right|{x}_{t},{y}_{t})\xb7p\left({y}_{t+\tau}\right|{x}_{t},{y}_{t},{x}_{t+\tau})=p({x}_{t},{y}_{t})\xb7p\left({y}_{t+\tau}\right|{x}_{t},{y}_{t})\xb7p\left({x}_{t+\tau}\right|{x}_{t},{y}_{t},{y}_{t+\tau})$.

**Figure 2.**Causal representation of signal-response models. The joint probability density is factorized into $p\left({\zeta}_{\tau}^{xy}\right)=p({x}_{t},{y}_{t})\xb7p\left({x}_{t+\tau}\right|{x}_{t})\xb7p\left({y}_{t+\tau}\right|{x}_{t},{y}_{t},{x}_{t+\tau})$.

**Figure 3.**Mapping irreversibility ${\Phi}_{\tau}^{xy}$, backward transfer entropy ${T}_{y\to x}(-\tau )$ and causal influence ${C}_{x\to y}\left(\tau \right)$ in the BLRM as a function of the observational time interval $\tau $. The parameters are $\beta =0.2$ and ${t}_{rel}=10$. All graphs are produced using R [41].

**Figure 4.**Mapping irreversibility density $\psi ({x}_{t},{y}_{t})$ for the BLRM at $\tau =0.5<\frac{1}{\beta}<{t}_{rel}$. The parameters are $\beta =0.2$ and ${t}_{rel}=10$. Both $\psi ({x}_{t},{y}_{t})$ and the space $(x,y)$ are expressed in units of standard deviations.

**Figure 5.**Mapping irreversibility density $\psi ({x}_{t},{y}_{t})$ for the BLRM at $\tau =25>{t}_{rel}>\frac{1}{\beta}$. The parameters are $\beta =0.2$ and ${t}_{rel}=10$. Both $\psi ({x}_{t},{y}_{t})$ and the space $(x,y)$ are expressed in units of standard deviations.

**Figure 6.**Mapping irreversibility and backward transfer entropy in our model of receptor-ligand systems (Equation (22)). The parameters are ${k}_{on}=5$, ${k}_{off}=1$, and $h=2$.

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Auconi, A.; Giansanti, A.; Klipp, E. Information Thermodynamics for Time Series of Signal-Response Models. *Entropy* **2019**, *21*, 177.
https://doi.org/10.3390/e21020177

**AMA Style**

Auconi A, Giansanti A, Klipp E. Information Thermodynamics for Time Series of Signal-Response Models. *Entropy*. 2019; 21(2):177.
https://doi.org/10.3390/e21020177

**Chicago/Turabian Style**

Auconi, Andrea, Andrea Giansanti, and Edda Klipp. 2019. "Information Thermodynamics for Time Series of Signal-Response Models" *Entropy* 21, no. 2: 177.
https://doi.org/10.3390/e21020177