Thermodynamics of Prediction

The thermodynamics of prediction Susanne Still, 1 David A. Sivak, 2 Anthony J. Bell, 3 and Gavin E. Crooks 2 University of Hawaii at M¯ anoa, Information and Computer Sciences, sstill@hawaii.edu Physical Biosciences Division, Lawrence Berkeley National Laboratory Redwood Center for Theoretical Neuroscience, University of California at Berkeley A system responding to a stochastic driving signal can be interpreted as computing, by means of its dynamics, an implicit model of the environmental variables. The system’s state retains information about past environmental fluctuations, and a fraction of this information is predictive of future ones. The remaining nonpredictive information reflects model complexity that does not improve predictive power, and thus represents the ineffectiveness of the model. We expose the fundamental equivalence between this model inefficiency and thermodynamic inefficiency, measured by the energy dissipated during the interaction between system and environment. Our results hold arbitrarily far from thermodynamic equilibrium and are applicable to a wide range of systems, including biomolecular machines. They highlight a profound connection between the effective use of information and efficient thermodynamic operation: any system constructed to keep memory about its environment and to operate with maximal energetic efficiency has to be predictive. All systems perform computations by means of re- sponding to their environment. In particular, living sys- tems compute, on a variety of length- and time-scales, fu- ture expectations based on their prior experience. Most biological computation is fundamentally a nonequilib- rium process, because a preponderance of biological ma- chinery in its natural operation is driven far from thermo- dynamic equilibrium. For example, many molecular ma- chines (such as the microtubule-associated motor kinesin) are driven by ATP hydrolysis, which liberates ∼500 meV per molecule [32]. This energy is large compared with ambient thermal energy, 1 k B T ≈ 25 meV (k B is Boltz- mann’s constant and the temperature is T ∼ 300 Kelvin). In general, such large energetic inputs drive the opera- tive degrees of freedom of biological machines away from equilibrium averages. Recently, significant progress has been made in de- scribing driven systems far from equilibrium [2], per- DISCLAIMER: This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. haps most notably Jarzynski’s work relation [33] gen- eralizing Clausius’ Inequality; the further generalization embodied in fluctuation theorems [34, 35]; and the ex- tension of these relations to calculating potentials of mean force [36]. These advances have allowed researchers to measure equilibrium quantities, such as free energy changes, by observing how a system reacts to being driven out of equilibrium, e.g. [39, 40]. This literature typically assumes that the experiment is known, i.e. that the exact time course of the driving signal is given. However, systems that are embedded in realistic environments, for example, a biological macro- molecule operating under natural conditions, are exposed to a stochastic drive. Here, we therefore study driven systems for which the changes in the driving signal(s) are governed by some probability density P X . This can be any stochastic process, and the results we derive re- quire neither that P X has specific properties, nor that it is known by the system. The dissipation, averaged not only over the system’s path through its state space, but also over driving protocols, then quantifies the system’s energetic inefficiency. We assume that there is no feed- back from the system to the driving signal. The system dynamics perform a computation by changing the system’s state, as a function of the driv- ing signal. As a result, the new system state contains some memory about the driving signal. The dynamics of the system can be interpreted as computing a model: past environmental influences are mapped onto the cur- rent state of the system, which through its correlation with forthcoming environmental fluctuations implicitly contains a prediction of the future. In this paper, we ask how the quality of this (implicit) model is related to thermodynamic inefficiency. But how do we measure the quality of a model? A useful model has to have predictive power (see e.g. [41–44], and refs.