Limits of predictions in thermodynamic systems: a review

Optimizing dynamical functions for speed with stochastic paths

Living systems are built from microscopic components that function dynamically; they generate work with molecular motors, assemble and disassemble structures such as microtubules, keep time with circadian clocks, and catalyze the replication of DNA. How do we implement these functions in synthetic nanostructured materials to execute them before the onset of dissipative losses? Answering this question requires a quantitative understanding of when we can improve performance and speed while minimizing the dissipative losses associated with operating in a fluctuating environment. Here, we show that there are four modalities for optimizing dynamical functions that can guide the design of nanoscale systems. We analyze Markov models that span the design space: a clock, ratchet, replicator, and self-assembling system. Using stochastic thermodynamics and an exact expression for path probabilities, we classify these models of dynamical functions based on the correlation of speed with dissipation and with the chosen performance metric. We also analyze random networks to identify the model features that affect their classification and the optimization of their functionality. Overall, our results show that the possible nonequilibrium paths can determine our ability to optimize the performance of dynamical functions, despite ever-present dissipation, when there is a need for speed.

Dynamic scaling of stochastic thermodynamic observables for chemical reactions at and away from equilibrium

Physical kinetic roughening processes are well-known to exhibit universal scaling of observables that fluctuate in space and time. Are there analogous dynamic scaling laws that are unique to the chemical reaction mechanisms available synthetically and occurring naturally? Here, we formulate an approach to the dynamic scaling of stochastic fluctuations in thermodynamic observables at and away from equilibrium. Both analytical expressions and numerical simulations confirm our dynamic scaling ansatz with associated scaling exponents, function, and law. A survey of common chemical mechanisms reveals classes that organize according to the molecularity of the reactions involved, the nature of the reaction vessel and external reservoirs, (non)equilibrium conditions, and the extent of autocatalysis in the reaction network. Varying experimental parameters, such as temperature, can cause coupled reactions capable of chemical feedback to transition between these classes. While path observables, such as the dynamical activity, have scaling exponents that are time-independent, the variance in the entropy production and flow can have time-dependent scaling exponents and self-averaging properties as a result of temporal correlations that emerge during thermodynamically irreversible processes. Altogether, these results establish dynamic universality classes in the nonequilibrium fluctuations of thermodynamic observables for well-mixed chemical reactions.

Stochastic paths controlling speed and dissipation

Natural processes occur in a finite amount of time and dissipate energy, entropy, and matter. Near equilibrium, thermodynamic intuition suggests that fast irreversible processes will dissipate more energy and entropy than slow quasistatic processes connecting the same initial and final states. For small systems, recently discovered thermodynamic speed limits suggest that faster processes will dissipate more than slower processes. Here, we test the hypothesis that this relationship between speed and dissipation holds for stochastic paths far from equilibrium. To analyze stochastic paths on finite timescales, we derive an exact expression for the path probabilities of continuous-time Markov chains from the path summation solution to the master equation. We present a minimal model for a driven system in which relative energies of the initial and target states control the speed, and the nonequilibrium currents of a cycle control the dissipation. Although the hypothesis holds near equilibrium, we find that faster processes can dissipate less under far-from-equilibrium conditions because of strong currents. This model serves as a minimal prototype for designing kinetics to sculpt the nonequilibrium path space so that faster paths produce less dissipation.

Hamiltonian active particles in an environment

We examine a Hamiltonian system which represents an active particle that can move against an opposing external force by drawing energy from an internal depot while immersed in a noisy and dissipative environment. The Hamiltonian consists of two subsystems, one representing the active particle's motion and the other its depot of "fuel." We show that although the active particle loses some of its energy to dissipation from the environment, dissipation can also help to stabilize the dynamical process that makes the particle active. This raises the possibility that the internal mechanisms of active particles may not only be able to operate in noisy and dissipative environments, but may actually rely on the environment for their control.

Undefining life's biochemistry: implications for abiogenesis

In the mid-twentieth century, multiple Nobel Prizes rewarded discoveries of a seemingly universal set of molecules and interactions that collectively defined the chemical basis for life. Twenty-first-century science knows that every detail of this Central Dogma of Molecular Biology can vary through either biological evolution, human engineering (synthetic biology) or both. Clearly the material, molecular basis of replicating, evolving entities can be different. There is far less clarity yet for what constitutes this set of possibilities. One approach to better understand the limits and scope of moving beyond life's central dogma comes from those who study life's origins. RNA, proteins and the genetic code that binds them each look like products of natural selection. This raises the question of what step(s) preceded these particular components? Answers here will clarify whether any discrete point in time or biochemical evolution will objectively merit the label of life's origin, or whether life unfolds seamlessly from the non-living universe.

Similarity of extremely rare nonequilibrium processes to equilibrium processes

For system coupled to heat baths, typical nonequilibrated processes, e.g., induced by varying an external parameter without waiting for equilibration in between, are very different from the corresponding equilibrium infinitely slow processes. Nevertheless, there are connections between equilibrium and nonequilibrated behaviors, e.g., the theorems of Jarzynski and Crooks, which relate the distribution P(W) of nonequilibrium work to the free energy differences ΔF. Here we study the naturally arising question, whether those relevant but rare trajectories, which exhibit these work values, show a higher degree of similarity to equilibrium. For convenience, we have chosen a simple model of RNA secondary structures (or single-stranded DNA), here modeling a medium-size hairpin structure, under influence of a varying external force. This allows us to measure the work W during the resulting fast unfolding and refolding processes within Monte Carlo simulations, i.e., in nonequilibrium. Also we sample numerically efficiently directly in exact equilibrium, for comparison. Using a sophisticated large-deviation algorithm, we are able to measure work distributions with high precision down to probabilities as small as 10^{-46}, enabling us to verify the Crooks and Jarzynski theorems. Furthermore, we analyze force-extension curves and the configurations of the secondary structures during unfolding and refolding for typical equilibrium processes and nonequilibrated processes. We find that the nonequilibrated processes where the work values are close to those which are most relevant for applying Crooks and Jarzynski theorems, respectively, but which occur with exponential small probabilities, are most and quite similar to the equilibrium processes.

Thermodynamic Optimality of Glycolytic Oscillations

Temporal order in living matters reflects the self-organizing nature of dynamical processes driven out of thermodynamic equilibrium. Because of functional reasons, the period of a biochemical oscillation must be tuned to a specific value with precision; however, according to the thermodynamic uncertainty relation (TUR), the precision of the oscillatory period is constrained by the thermodynamic cost of generating it. After reviewing the basics of chemical oscillations using the Brusselator as a model system, we study the glycolytic oscillation generated by octameric phosphofructokinase (PFK), which is known to display a period of several minutes. By exploring the phase space of glycolytic oscillations, we find that the glycolytic oscillation under the cellular condition is realized in a cost-effective manner. Specifically, over the biologically relevant range of parameter values of glycolysis and octameric PFK, the entropy production from the glycolytic oscillation is minimal when the oscillation period is (5-10) min. Furthermore, the glycolytic oscillation is found at work near the phase boundary of limit cycles, suggesting that a moderate increase of glucose injection rate leads to the loss of oscillatory dynamics, which is reminiscent of the loss of pulsatile insulin release resulting from elevated blood glucose level.

Detailed Balance Broken by Catch Bond Kinetics Enables Mechanical-Adaptation in Active Materials

Unlike nearly all engineered materials which contain bonds that weaken under load, biological materials contain "catch" bonds which are reinforced under load. Consequently, materials, such as the cell cytoskeleton, can adapt their mechanical properties in response to their state of internal, non-equilibrium (active) stress. However, how large-scale material properties vary with the distance from equilibrium is unknown, as are the relative roles of active stress and binding kinetics in establishing this distance. Through course-grained molecular dynamics simulations, the effect of breaking of detailed balance by catch bonds on the accumulation and dissipation of energy within a model of the actomyosin cytoskeleton is explored. It is found that the extent to which detailed balance is broken uniquely determines a large-scale fluid-solid transition with characteristic time-reversal symmetries. The transition depends critically on the strength of the catch bond, suggesting that active stress is necessary but insufficient to mount an adaptive mechanical response.

A Darwinian and Physical Look at Stem Cell Biology Helps Understanding the Role of Stochasticity in Development

Single-cell analysis allows biologists to gain huge insight into cell differentiation and tissue structuration. Randomness of differentiation, both in vitro and in vivo, of pluripotent (multipotent) stem cells is now demonstrated to be mainly based on stochastic gene expression. Nevertheless, it remains necessary to incorporate this inherent stochasticity of developmental processes within a coherent scheme. We argue here that the theory called ontophylogenesis is more relevant and better fits with experimental data than alternative theories which have been suggested based on the notions of self-organization and attractor states. The ontophylogenesis theory considers the generation of a differentiated state as a constrained random process: randomness is provided by the stochastic dynamics of biochemical reactions while the environmental constraints, including cell inner structures and cell-cell interactions, drive the system toward a stabilized state of equilibrium. In this conception, biological organization during development can be seen as the result of multiscale constraints produced by the dynamical organization of the biological system which retroacts on the stochastic dynamics at lower scales. This scheme makes it possible to really understand how the generation of reproducible structures at higher organization levels can be fully compatible with probabilistic behavior at the lower levels. It is compatible with the second law of thermodynamics but allows the overtaking of the limitations exhibited by models only based on entropy exchanges which cannot cope with the description nor the dynamics of the mesoscopic and macroscopic organization of biological systems.

The thermodynamic uncertainty relation in biochemical oscillations

Living systems regulate many aspects of their behaviour through periodic oscillations of molecular concentrations, which function as 'biochemical clocks.' The chemical reactions that drive these clocks are intrinsically stochastic at the molecular level, so that the duration of a full oscillation cycle is subject to random fluctuations. Their success in carrying out their biological function is thought to depend on the degree to which these fluctuations in the cycle period can be suppressed. Biochemical oscillators also require a constant supply of free energy in order to break detailed balance and maintain their cyclic dynamics. For a given free energy budget, the recently discovered 'thermodynamic uncertainty relation' yields the magnitude of period fluctuations in the most precise conceivable free-running clock. In this paper, we show that computational models of real biochemical clocks severely underperform this optimum, with fluctuations several orders of magnitude larger than the theoretical minimum. We argue that this suboptimal performance is due to the small number of internal states per molecule in these models, combined with the high level of thermodynamic force required to maintain the system in the oscillatory phase. We introduce a new model with a tunable number of internal states per molecule and confirm that it approaches the optimal precision as this number increases.

Energy storage in steady states under cyclic local energy input

We study periodic steady states of a lattice system under external cyclic energy supply using simulation. We consider different protocols for cyclic energy supply and examine the energy storage. Under the same energy flux, we found that the stored energy depends on the details of the supply, period, and amplitude of the supply. Further, we introduce an adiabatic wall as an internal constraint into the lattice and examine the stored energy with respect to different positions of the internal constrain. We found that the stored energy for constrained systems is larger than its unconstrained counterpart. We also observe that the system stores more energy through large and rare energy delivery, comparing to small and frequent delivery.

Systems protobiology: origin of life in lipid catalytic networks

Life is that which replicates and evolves, but there is no consensus on how life emerged. We advocate a systems protobiology view, whereby the first replicators were assemblies of spontaneously accreting, heterogeneous and mostly non-canonical amphiphiles. This view is substantiated by rigorous chemical kinetics simulations of the graded autocatalysis replication domain (GARD) model, based on the notion that the replication or reproduction of compositional information predated that of sequence information. GARD reveals the emergence of privileged non-equilibrium assemblies (composomes), which portray catalysis-based homeostatic (concentration-preserving) growth. Such a process, along with occasional assembly fission, embodies cell-like reproduction. GARD pre-RNA evolution is evidenced in the selection of different composomes within a sparse fitness landscape, in response to environmental chemical changes. These observations refute claims that GARD assemblies (or other mutually catalytic networks in the metabolism first scenario) cannot evolve. Composomes represent both a genotype and a selectable phenotype, anteceding present-day biology in which the two are mostly separated. Detailed GARD analyses show attractor-like transitions from random assemblies to self-organized composomes, with negative entropy change, thus establishing composomes as dissipative systems—hallmarks of life. We show a preliminary new version of our model, metabolic GARD (M-GARD), in which lipid covalent modifications are orchestrated by non-enzymatic lipid catalysts, themselves compositionally reproduced. M-GARD fills the gap of the lack of true metabolism in basic GARD, and is rewardingly supported by a published experimental instance of a lipid-based mutually catalytic network. Anticipating near-future far-reaching progress of molecular dynamics, M-GARD is slated to quantitatively depict elaborate protocells, with orchestrated reproduction of both lipid bilayer and lumenal content. Finally, a GARD analysis in a whole-planet context offers the potential for estimating the probability of life's emergence. The invigorated GARD scrutiny presented in this review enhances the validity of autocatalytic sets as a bona fide early evolution scenario and provides essential infrastructure for a paradigm shift towards a systems protobiology view of life's origin.

The Energy Homeostasis Principle: Neuronal Energy Regulation Drives Local Network Dynamics Generating Behavior

A major goal of neuroscience is understanding how neurons arrange themselves into neural networks that result in behavior. Most theoretical and experimental efforts have focused on a top-down approach which seeks to identify neuronal correlates of behaviors. This has been accomplished by effectively mapping specific behaviors to distinct neural patterns, or by creating computational models that produce a desired behavioral outcome. Nonetheless, these approaches have only implicitly considered the fact that neural tissue, like any other physical system, is subjected to several restrictions and boundaries of operations. Here, we proposed a new, bottom-up conceptual paradigm: The Energy Homeostasis Principle, where the balance between energy income, expenditure, and availability are the key parameters in determining the dynamics of neuronal phenomena found from molecular to behavioral levels. Neurons display high energy consumption relative to other cells, with metabolic consumption of the brain representing 20% of the whole-body oxygen uptake, contrasting with this organ representing only 2% of the body weight. Also, neurons have specialized surrounding tissue providing the necessary energy which, in the case of the brain, is provided by astrocytes. Moreover, and unlike other cell types with high energy demands such as muscle cells, neurons have strict aerobic metabolism. These facts indicate that neurons are highly sensitive to energy limitations, with Gibb's free energy dictating the direction of all cellular metabolic processes. From this activity, the largest energy, by far, is expended by action potentials and post-synaptic potentials; therefore, plasticity can be reinterpreted in terms of their energy context. Consequently, neurons, through their synapses, impose energy demands over post-synaptic neurons in a close loop-manner, modulating the dynamics of local circuits. Subsequently, the energy dynamics end up impacting the homeostatic mechanisms of neuronal networks. Furthermore, local energy management also emerges as a neural population property, where most of the energy expenses are triggered by sensory or other modulatory inputs. Local energy management in neurons may be sufficient to explain the emergence of behavior, enabling the assessment of which properties arise in neural circuits and how. Essentially, the proposal of the Energy Homeostasis Principle is also readily testable for simple neuronal networks.

Active regeneration unites high- and low-temperature features in cooperative self-assembly

Cytoskeletal filaments are capable of self-assembly in the absence of externally supplied chemical energy, but the rapid turnover rates essential for their biological function require a constant flux of adenosine triphosphate (ATP) or guanosine triphosphate (GTP) hydrolysis. The same is true for two-dimensional protein assemblies employed in the formation of vesicles from cellular membranes, which rely on ATP-hydrolyzing enzymes to rapidly disassemble upon completion of the process. Recent observations suggest that the nucleolus, p granules, and other three-dimensional membraneless organelles may also demand dissipation of chemical energy to maintain their fluidity. Cooperative binding plays a crucial role in the dynamics of these higher-dimensional structures, but is absent from classic models of one-dimensional cytoskeletal assembly. In this paper, we present a thermodynamically consistent model of active regeneration with cooperative assembly, and compute the maximum turnover rate and minimum disassembly time as a function of the chemical driving force and the binding energy. We find that these driven structures resemble different equilibrium states above and below the nucleation barrier. In particular, we show that the maximal acceleration under large binding energies unites infinite-temperature local fluctuations with low-temperature nucleation kinetics.