Energy-speed-accuracy relation in complex networks for biological discrimination

Engineering TCR-controlled fuzzy logic into CAR T cells enhances therapeutic specificity

Chimeric antigen receptor (CAR) T cell immunotherapy represents a breakthrough in the treatment of hematological malignancies, but poor specificity has limited its applicability to solid tumors. By contrast, natural T cells harboring T cell receptors (TCRs) can discriminate between neoantigen-expressing cancer cells and self-antigen-expressing healthy tissues but have limited potency against tumors. We used a high-throughput platform to systematically evaluate the impact of co-expressing a TCR and CAR on the same CAR T cell. While strong TCR-antigen interactions enhanced CAR activation, weak TCR-antigen interactions actively antagonized their activation. Mathematical modeling captured this TCR-CAR crosstalk in CAR T cells, allowing us to engineer dual TCR/CAR T cells targeting neoantigens (HHATL8F/p53R175H) and human epithelial growth factor receptor 2 (HER2) ligands, respectively. These T cells exhibited superior anti-cancer activity and minimal toxicity against healthy tissue compared with conventional CAR T cells in a humanized solid tumor mouse model. Harnessing pre-existing inhibitory crosstalk between receptors, therefore, paves the way for the design of more precise cancer immunotherapies.

Speed-Energy-Efficiency Trade-off in Hsp70 Chaperone System

Proteins must fold into their native structure to carry out cellular functions. However, they can sometimes misfold into non-native structures, leading to reduced efficiency or malfunction. Chaperones help prevent misfolding by guiding proteins to their active state using energy from ATP hydrolysis. Experiments have revealed numerous kinetic and structural aspects of how various chaperones facilitate the folding of proteins into their native structure. However, what remains missing is a fundamental theoretical understanding of their operational mechanisms, especially the limits and constraints imposed on their efficiency by energy flow and dissipation. To address this, we built a kinetic model of the Hsp70 chaperone system by incorporating all key structural and kinetic details. Then, using the chemical kinetic equations, we investigate how energy expenditure shapes the efficiency of Hsp70 chaperones in the proper folding of misfolded proteins. We show that ATP consumption by chaperones significantly enhances the folding of proteins into their native states. Our investigations reveal that a chaperone achieves optimal efficiency when its binding to misfolded proteins is much faster than the misfolding kinetics of that protein. We also demonstrate the presence of an upper bound on a chaperone's efficiency of protein folding and its overall rescue rate. This upper bound increases with energy dissipation until it reaches a saturation point. Furthermore, we show a speed-energy-efficiency trade-off in chaperone action, demonstrating that it is impossible to simultaneously optimize the efficiency of chaperone-assisted protein folding and the energy efficiency of the process.

Topologically constrained fluctuations and thermodynamics regulate nonequilibrium response

The limits on a system's response to external perturbations inform our understanding of how physical properties can be shaped by microscopic characteristics. Here, we derive constraints on the steady-state nonequilibrium response of physical observables in terms of the topology of the microscopic state space and the strength of thermodynamic driving. Notably, evaluation of these limits requires no kinetic information beyond the state-space structure. When applied to models of receptor binding, we find that sensitivity is bounded by the steepness of a Hill function with a Hill coefficient enhanced by the chemical driving beyond the structural equilibrium limit.

A universal method for analyzing copolymer growth

Polymers consisting of more than one type of monomer, known as copolymers, are vital to both living and synthetic systems. Copolymerization has been studied theoretically in a number of contexts, often by considering a Markov process in which monomers are added or removed from the growing tip of a long copolymer. To date, the analysis of the most general models of this class has necessitated simulation. We present a general method for analyzing such processes without resorting to simulation. Our method can be applied to models with an arbitrary network of sub-steps prior to addition or removal of a monomer, including non-equilibrium kinetic proofreading cycles. Moreover, the approach allows for a dependency of addition and removal reactions on the neighboring site in the copolymer and thermodynamically self-consistent models in which all steps are assumed to be microscopically reversible. Using our approach, thermodynamic quantities such as chemical work; kinetic quantities such as time taken to grow; and statistical quantities such as the distribution of monomer types in the growing copolymer can be directly derived either analytically or numerically from the model definition.

Reformulating nonequilibrium steady states and generalized Hopfield discrimination

Despite substantial progress in nonequilibrium physics, steady-state (s.s.) probabilities remain intractable to analysis. For a Markov process, s.s. probabilities can be expressed in terms of transition rates using the graph-theoretic matrix-tree theorem (MTT). The MTT reveals that, away from equilibrium, s.s. probabilities become globally dependent on all rates, with expressions growing exponentially in the system size. This overwhelming complexity and lack of thermodynamic interpretation have greatly impeded analysis. Here we show that s.s. probabilities are proportional to the average of exp(-S(P)), where S(P) is the entropy generated along minimal paths P in the graph, and the average is taken over a probability distribution on spanning trees. Assuming Arrhenius rates, this "arboreal" distribution becomes Boltzmann-like, with the energy of a tree being its total edge barrier energy. This reformulation offers a thermodynamic interpretation that smoothly generalizes equilibrium statistical mechanics and reorganizes the expression complexity: the number of distinct minimal-path entropies depends not on graph size but on the entropy production index, a measure of nonequilibrium complexity. We demonstrate the power of this reformulation by extending Hopfield's analysis of discrimination by kinetic proofreading to any graph with index 1. We derive a general formula for the error ratio and use it to show how local energy dissipation can yield optimal discrimination through global synergy.

The linear framework: using graph theory to reveal the algebra and thermodynamics of biomolecular systems

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent biochemical species or molecular states, edges represent reactions or transitions and labels represent rates. The graph yields a linear dynamics for molecular concentrations or state probabilities, with the graph Laplacian as the operator, and the labels encode the nonlinear interactions between system and environment. The labels can be specified by vertices of other graphs or by conservation laws or, when the environment consists of thermodynamic reservoirs, they may be constants. In the latter case, the graphs correspond to infinitesimal generators of Markov processes. The key advantage of the framework has been that steady states are determined as rational algebraic functions of the labels by the Matrix-Tree theorems of graph theory. When the system is at thermodynamic equilibrium, this prescription recovers equilibrium statistical mechanics but it continues to hold for non-equilibrium steady states. The framework goes beyond other graph-based approaches in treating the graph as a mathematical object, for which general theorems can be formulated that accommodate biomolecular complexity. It has been particularly effective at analysing enzyme-catalysed modification systems and input-output responses.

The optimal strategy balancing risk and speed predicts DNA damage checkpoint override times

Checkpoints arrest biological processes allowing time for error correction. The phenomenon of checkpoint override (also known as checkpoint adaptation, slippage, or leakage), during cellular self-replication is biologically critical but currently lacks a quantitative, functional, or system-level understanding. To uncover fundamental laws governing error-correction systems, we derived a general theory of optimal checkpoint strategies, balancing the trade-off between risk and self-replication speed. Mathematically, the problem maps onto the optimization of an absorbing boundary for a random walk. We applied the theory to the DNA damage checkpoint (DDC) in budding yeast, an intensively researched model checkpoint. Using novel reporters for double-strand DNA breaks (DSBs), we first quantified the probability distribution of DSB repair in time including rare events and, secondly, the survival probability after override. With these inputs, the optimal theory predicted remarkably accurately override times as a function of DSB numbers, which we measured precisely for the first time. Thus, a first-principles calculation revealed undiscovered patterns underlying highly noisy override processes. Our multi-DSB measurements revise well-known past results and show that override is more general than previously thought.

Kinetic Proofreading

Biochemistry and molecular biology rely on the recognition of structural complementarity between molecules. Molecular interactions must be both quickly reversible, i.e., tenuous, and specific. How the cell reconciles these conflicting demands is the subject of this article. The problem and its theoretical solution are discussed within the wider theoretical context of the thermodynamics of stochastic processes (stochastic thermodynamics). The solution-an irreversible reaction cycle that decreases internal error at the expense of entropy export into the environment-is shown to be widely employed by biological processes that transmit genetic and regulatory information. Expected final online publication date for the Annual Review of Biochemistry, Volume 91 is June 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

The energy cost and optimal design of networks for biological discrimination

Many biological processes discriminate between correct and incorrect substrates through the kinetic proofreading mechanism that enables lower error at the cost of higher energy dissipation. Elucidating physico-chemical constraints for global minimization of dissipation and error is important for understanding enzyme evolution. Here, we identify theoretically a fundamental error-cost bound that tightly constrains the performance of proofreading networks under any parameter variations preserving the rate discrimination between substrates. The bound is kinetically controlled, i.e. completely determined by the difference between the transition state energies on the underlying free energy landscape. The importance of the bound is analysed for three biological processes. DNA replication by T7 DNA polymerase is shown to be nearly optimized, i.e. its kinetic parameters place it in the immediate proximity of the error-cost bound. The isoleucyl-tRNA synthetase (IleRS) of E. coli also operates close to the bound, but further optimization is prevented by the need for reaction speed. In contrast, E. coli ribosome operates in a high-dissipation regime, potentially in order to speed up protein production. Together, these findings establish a fundamental error-dissipation relation in biological proofreading networks and provide a theoretical framework for studying error-dissipation trade-off in other systems with biological discrimination.

Noise-induced quasiperiod and period switching

We employ a typical genetic circuit model to explore how noise can influence dynamic structure. With the increase of a key interactive parameter, the model will deterministically go through two bifurcations and three dynamic structure regions. We find that a quasiperiodic component, which is not allowed by deterministic dynamics, will be generated by noise inducing in the first two regions, and this quasiperiod will be more and more stable along with the increase in noise. In particular, in the second region the quasiperiod will compete with a stable limit cycle and perform a new transient rhythm. Furthermore, we ascertain the entropy production rate and the heat dissipation rate, and discover a minimal value with theoretical elucidation. In the end, we unveil the mechanism of the formation of quasiperiods, and show a practical biological example. We expect this work to be helpful in solving some biological or ecological problems, such as the genetic origin of periodical cicadas and population dynamics with fluctuation.

Allosteric conformational ensembles have unlimited capacity for integrating information

Integration of binding information by macromolecular entities is fundamental to cellular functionality. Recent work has shown that such integration cannot be explained by pairwise cooperativities, in which binding is modulated by binding at another site. Higher-order cooperativities (HOCs), in which binding is collectively modulated by multiple other binding events, appear to be necessary but an appropriate mechanism has been lacking. We show here that HOCs arise through allostery, in which effective cooperativity emerges indirectly from an ensemble of dynamically interchanging conformations. Conformational ensembles play important roles in many cellular processes but their integrative capabilities remain poorly understood. We show that sufficiently complex ensembles can implement any form of information integration achievable without energy expenditure, including all patterns of HOCs. Our results provide a rigorous biophysical foundation for analysing the integration of binding information through allostery. We discuss the implications for eukaryotic gene regulation, where complex conformational dynamics accompanies widespread information integration.

Thermodynamic uncertainty relation to assess biological processes

We review the trade-offs between speed, fluctuations, and thermodynamic cost involved with biological processes in nonequilibrium states and discuss how optimal these processes are in light of the universal bound set by the thermodynamic uncertainty relation (TUR). The values of the uncertainty product Q of TUR, which can be used as a measure of the precision of enzymatic processes realized for a given thermodynamic cost, are suboptimal when the substrate concentration is at the Michaelis constant, and some of the key biological processes are found to work around this condition. We illustrate the utility of Q in assessing how close the molecular motors and biomass producing machineries are to the TUR bound, and for the cases of biomass production (or biological copying processes), we discuss how their optimality quantified in terms of Q is balanced with the error rate in the information transfer process. We also touch upon the trade-offs in other error-minimizing processes in biology, such as gene regulation and chaperone-assisted protein folding. A spectrum of Q recapitulating the biological processes surveyed here provides glimpses into how biological systems are evolved to optimize and balance the conflicting functional requirements.

Theoretical Analysis Reveals the Cost and Benefit of Proofreading in Coronavirus Genome Replication

Severe acute respiratory syndrome coronaviruses have unusually large RNA genomes replicated by a multiprotein complex containing an RNA-dependent RNA polymerase (RdRp). Exonuclease activity enables the RdRp complex to remove wrongly incorporated bases via proofreading, a process not utilized by other RNA viruses. However, it is unclear why the RdRp complex needs proofreading and what the associated trade-offs are. Here we investigate the interplay among the accuracy, speed, and energetic cost of proofreading in the RdRp complex using a kinetic model and bioinformatics analysis. We find that proofreading nearly optimizes the rate of functional virus production. However, we find that further optimization would lead to a significant increase in the proofreading cost. Unexpected importance of the cost minimization is further supported by other global analyses. We speculate that cost optimization could help avoid cell defense responses. Thus, proofreading is essential for the production of functional viruses, but its rate is limited by energy costs.

Role of thermal fluctuations in biological copying mechanisms

During transcription, translation, or self-replication of DNA or RNA, information is transferred to the newly formed species from its predecessor. These processes can be interpreted as (generalized) biological copying mechanism as the new biological entities like DNA, RNA, or proteins are representing the information of their parent bodies uniquely. The accuracy of these copying processes is essential, since errors in the copied code can reduce the functionality of the next generation. Such errors might result from perturbations on these processes. Most important in this context is the temperature of the medium, i.e., thermal noise. Although a reasonable amount of experimental studies have been carried out on this important issue, theoretical understanding is truly sparse. In the present work, we illustrate a model study which is able to focus on the effect of the temperature on the process of biological copying mechanisms, as well as on mutation. We find for our paradigmatic models that, in a quite general scenario, the copying processes are most accurate at an intermediate temperature range; i.e., there exists an optimum temperature where mutation is most unlikely. This allows us to interpret the observations for some biological species with the aid of our model study.

Proofreading through spatial gradients

Key enzymatic processes use the nonequilibrium error correction mechanism called kinetic proofreading to enhance their specificity. The applicability of traditional proofreading schemes, however, is limited because they typically require dedicated structural features in the enzyme, such as a nucleotide hydrolysis site or multiple intermediate conformations. Here, we explore an alternative conceptual mechanism that achieves error correction by having substrate binding and subsequent product formation occur at distinct physical locations. The time taken by the enzyme–substrate complex to diffuse from one location to another is leveraged to discard wrong substrates. This mechanism does not have the typical structural requirements, making it easier to overlook in experiments. We discuss how the length scales of molecular gradients dictate proofreading performance, and quantify the limitations imposed by realistic diffusion and reaction rates. Our work broadens the applicability of kinetic proofreading and sets the stage for studying spatial gradients as a possible route to specificity.

Gene Regulation in and out of Equilibrium

Determining whether and how a gene is transcribed are two of the central processes of life. The conceptual basis for understanding such gene regulation arose from pioneering biophysical studies in eubacteria. However, eukaryotic genomes exhibit vastly greater complexity, which raises questions not addressed by this bacterial paradigm. First, how is information integrated from many widely separated binding sites to determine how a gene is transcribed? Second, does the presence of multiple energy-expending mechanisms, which are absent from eubacterial genomes, indicate that eukaryotes are capable of improved forms of genetic information processing? An updated biophysical foundation is needed to answer such questions. We describe the linear framework, a graph-based approach to Markov processes, and show that it can accommodate many previous studies in the field. Under the assumption of thermodynamic equilibrium, we introduce a language of higher-order cooperativities and show how it can rigorously quantify gene regulatory properties suggested by experiment. We point out that fundamental limits to information processing arise at thermodynamic equilibrium and can only be bypassed through energy expenditure. Finally, we outline some of the mathematical challenges that must be overcome to construct an improved biophysical understanding of gene regulation.

Do We Understand the Mechanisms Used by Biological Systems to Correct Their Errors?

Most cellular processes involved in biological information processing display a surprisingly low error rate despite the stochasticity of the underlying biochemical reactions and the presence of competing chemical species. Such high fidelity is the result of nonequilibrium kinetic proofreading mechanisms, i.e., the existence of dissipative pathways for correcting the reactions that went in the wrong direction. While proofreading was often studied from the perspective of error minimization, a number of recent studies have demonstrated that the underlying mechanisms need to consider the interplay of other characteristic properties such as speed, energy dissipation, and noise reduction. Here, we present current views and new insights on the mechanisms of error-correction phenomena and various trade-off scenarios in the optimization of the functionality of biological systems. Existing challenges and future directions are also discussed.

Reversal symmetries for cyclic paths away from thermodynamic equilibrium

If a system is at thermodynamic equilibrium, an observer cannot tell whether a film of it is being played forward or in reverse: any transition will occur with the same frequency in the forward as in the reverse direction. However, if expenditure of energy changes the rate of even a single transition to yield a nonequilibrium steady state, such time-reversal symmetry undergoes a widespread breakdown, far beyond the point at which the energy is expended. An explosion of interdependency also arises, with steady-state probabilities of system states depending in a complicated manner on the rate of every transition in the system. Nevertheless, in the midst of this global nonequilibrium complexity, we find that cyclic paths have reversibility properties that remain local, and which can exhibit symmetry, no matter how far the system is from thermodynamic equilibrium. Specifically, given any cycle of reversible transitions, the ratio of the frequencies with which the cycle occurs in one direction versus the other is determined, in the long-time limit, only by the thermodynamic force on the cycle itself, without requiring knowledge of transition rates elsewhere in the system. In particular, if there is no net energy expenditure on the cycle, then, over long times, the cycle occurrence frequencies are the same in either direction.

Trade-Offs between Speed, Accuracy, and Dissipation in tRNAIle Aminoacylation

Living systems maintain a high fidelity in information processing through kinetic proofreading, a mechanism for preferentially removing incorrect substrates at the cost of energy dissipation and slower speed. Proofreading mechanisms must balance their demand for higher speed, fewer errors, and lower dissipation, but it is unclear how rates of individual reaction steps are evolutionarily tuned to balance these needs, especially when multiple proofreading mechanisms are present. Here, using a discrete-state stochastic model, we analyze the optimization strategies in Escherichia coli isoleucyl-tRNA synthetase. Surprisingly, this enzyme adopts an economic proofreading strategy and improves speed and dissipation as long as the error is tolerable. Through global parameter sampling, we reveal a fundamental dissipation-error relation that bounds the enzyme's optimal performance and explains the importance of the post-transfer editing mechanism. The proximity of native system parameters to this bound demonstrates the importance of energy dissipation as an evolutionary force affecting fitness.

Kinetic proofreading and the limits of thermodynamic uncertainty

To mitigate errors induced by the cell's heterogeneous noisy environment, its main information channels and production networks utilize the kinetic proofreading (KPR) mechanism. Here, we examine two extensively studied KPR circuits, DNA replication by the T7 DNA polymerase and translation by the E. coli ribosome. Using experimental data, we analyze the performance of these two vital systems in light of the fundamental bounds set by the recently discovered thermodynamic uncertainty relation (TUR), which places an inherent trade-off between the precision of a desirable output and the amount of energy dissipation required. We show that the DNA polymerase operates close to the TUR lower bound, while the ribosome operates ∼5 times farther from this bound. This difference originates from the enhanced binding discrimination of the polymerase which allows it to operate effectively as a reduced reaction cycle prioritizing correct product formation. We show that approaching this limit also decouples the thermodynamic uncertainty factor from speed and error, thereby relaxing the accuracy-speed trade-off of the system. Altogether, our results show that operating near this reduced cycle limit not only minimizes thermodynamic uncertainty, but also results in global performance enhancement of KPR circuits.

Allostery and Kinetic Proofreading

Kinetic proofreading is an error correction mechanism present in the processes of the central dogma and beyond and typically requires the free energy of nucleotide hydrolysis for its operation. Though the molecular players of many biological proofreading schemes are known, our understanding of how energy consumption is managed to promote fidelity remains incomplete. In our work, we introduce an alternative conceptual scheme called "the piston model of proofreading" in which enzyme activation through hydrolysis is replaced with allosteric activation achieved through mechanical work performed by a piston on regulatory ligands. Inspired by Feynman's ratchet and pawl mechanism, we consider a mechanical engine designed to drive the piston actions powered by a lowering weight, whose function is analogous to that of ATP synthase in cells. Thanks to its mechanical design, the piston model allows us to tune the "knobs" of the driving engine and probe the graded changes and trade-offs between speed, fidelity, and energy dissipation. It provides an intuitive explanation of the conditions necessary for optimal proofreading and reveals the unexpected capability of allosteric molecules to beat the Hopfield limit of fidelity by leveraging the diversity of states available to them. The framework that we have built for the piston model can also serve as a basis for additional studies of driven biochemical systems.

Precision in a rush: Trade-offs between reproducibility and steepness of the hunchback expression pattern

Fly development amazes us by the precision and reproducibility of gene expression, especially since the initial expression patterns are established during very short nuclear cycles. Recent live imaging of hunchback promoter dynamics shows a stable steep binary expression pattern established within the three minute interphase of nuclear cycle 11. Considering expression models of different complexity, we explore the trade-off between the ability of a regulatory system to produce a steep boundary and minimize expression variability between different nuclei. We show how a limited readout time imposed by short developmental cycles affects the gene’s ability to read positional information along the embryo’s anterior posterior axis and express reliably. Comparing our theoretical results to real-time monitoring of the hunchback transcription dynamics in live flies, we discuss possible regulatory strategies, suggesting an important role for additional binding sites, gradients or non-equilibrium binding and modified transcription factor search strategies.

Despite very limited time, organisms develop in reproducible ways. In the early stages of fly development the information about maternal signals is read out in a few minutes to produce steep and precise gene expression patterns. Motivated by recent live imaging experiments in fly embryos, we explore the consequences of the trade-off between a rushed but reproducible readout and a steep expression pattern on the regulatory modules of gene expression. We show that the current view of one anterior gradient morphogen binding to six binding sites is quantitatively inconsistent with the experimental data given the short readout time, suggesting other regulatory features.

Structural conditions on complex networks for the Michaelis-Menten input-output response

The Michaelis-Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input-output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a "coarse graining" into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula's ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.