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

Memory, hysteresis, and kinetic cooperativity in stochastic mnemonic networks

Mnemonic networks are cyclic catalytic networks of monomeric enzymes that exhibit kinetic cooperativity as departures of the mean velocity from the hyperbolic, Michaelis-Menten-like response. In addition, such networks admit a hysteretic response when conformational fluctuations are slow compared to the catalytic rate. Here, we show how these fluctuation-driven effects emerge from the underlying stochasticity in the network. We use the chemical master equation to study the stochastic kinetics of mnemonic networks, which, in their minimal form, include a pair of conformers and triangular reaction pathways. We introduce statistical measures that are conditional on the turnovers to comprehensively analyze molecular fluctuations in the transient and stationary states of these networks. In the transient state, temporal correlations between enzyme turnovers lead to an inequivalence between number and temporal fluctuations, yielding a hysteretic response of the mean velocity to substrates. The transient relaxes to a stationary state with independent and identically distributed turnovers and equality between number and temporal fluctuations. This state is a non-equilibrium stationary state (NESS) when the Kolmogorov loop criterion is not satisfied, leading to the emergence of kinetic cooperativity. The symmetry of the number correlation functions allows us to distinguish between the absence of cooperativity in equilibrium and the accidental vanishing of cooperativity in a NESS. We conclude that memory and hysteresis are transient effects while kinetic cooperativity emerges as the macroscopic manifestation of the microscopic irreversibility of the NESS in a network with cyclic reaction pathways.

Steady-state solution of Markov jump processes in terms of arrival probabilities

Several dynamical processes can be modeled as Markov jump processes among a finite number N of sites (the distinct physical states). Here we consider strongly connected networks with time-independent site-to-site jump rate constants, and focus on the steady-state occupation probabilities of the sites. We provide a physically framed expression of the steady-state distribution in terms of arrival probabilities, here defined as the probabilities of going from starting sites to target sites with a given number of jumps (regardless of the time required). In particular, the full set of return probabilities (for all the sites of the network) up to N-1 jumps is necessary and sufficient. A few examples illustrate the outcomes, including the case of stochastic chemical kinetics.

Steady-state probabilities for Markov jump processes in terms of powers of the transition rate matrix

Several types of dynamics at stationarity can be described in terms of a Markov jump process among a finite number N of representative sites. Before dealing with the dynamical aspects, one basic problem consists in expressing the a priori steady-state occupation probabilities of the sites. In particular, one wishes to go beyond the mere black-box computational tools and find expressions in which the jump rate constants appear explicitly, therefore allowing for a potential design/control of the network. For strongly connected networks admitting a unique stationary state with all sites populated, here we express the occupation probabilities in terms of a formula that involves powers of the transition rate matrix up to order N - 1. We also provide an expression of the derivatives with respect to the jump rate constants, possibly useful in sensitivity analysis frameworks. Although we refer to dynamics in (bio)chemical networks at thermal equilibrium or under nonequilibrium steady-state conditions, the results are valid for any Markov jump process under the same assumptions.

The Hill function is the universal Hopfield barrier for sharpness of input-output responses

The Hill functions, [Formula: see text], have been widely used in biology for over a century but, with the exception of [Formula: see text], they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input-output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, coregulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalizes most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input-output responses fall within an effectively bounded region of the positive quadrant, [Formula: see text], for any equilibrium model with [Formula: see text] input binding sites. [Formula: see text] exhibits a cusp which approaches, but never exceeds, the sharpness of [Formula: see text], but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers, and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, [Formula: see text], whose structure may be of mathematical interest, and suggest the importance of characterizing Hopfield barriers for other forms of cellular information processing.

The Hill function is the universal Hopfield barrier for sharpness of input-output responses

The Hill functions, ℋ h ( x ) = x h / 1 + x h , have been widely used in biology for over a century but, with the exception of ℋ 1 , they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input-output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, co-regulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalises most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input-output responses fall within an effectively bounded region of the positive quadrant, Ω m ⊂ ℝ + 2 , for any equilibrium model with m input binding sites. Ω m exhibits a cusp which approaches, but never exceeds, the sharpness of ℋ m but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, Ω m , whose structure may be of mathematical interest, and suggest the importance of characterising Hopfield barriers for other forms of cellular information processing.

The linear framework II: using graph theory to analyse the transient regime of Markov processes

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent chemical species or molecular states, edges represent reactions or transitions and edge labels represent rates that also describe how the system is interacting with its environment. The present paper is a sequel to a recent review of the framework that focussed on how graph-theoretic methods give insight into steady states as rational algebraic functions of the edge labels. Here, we focus on the transient regime for systems that correspond to continuous-time Markov processes. In this case, the graph specifies the infinitesimal generator of the process. We show how the moments of the first-passage time distribution, and related quantities, such as splitting probabilities and conditional first-passage times, can also be expressed as rational algebraic functions of the labels. This capability is timely, as new experimental methods are finally giving access to the transient dynamic regime and revealing the computations and information processing that occur before a steady state is reached. We illustrate the concepts, methods and formulas through examples and show how the results may be used to illuminate previous findings in the literature.

Molecular noise-induced activator-inhibitor duality in enzyme inhibition kinetics

Classical theories of enzyme inhibition kinetics predict a monotonic decrease in the mean catalytic activity with the increase in inhibitor concentration. The steady-state result, derived from deterministic mass action kinetics, ignores molecular noise in enzyme-inhibition mechanisms. Here, we present a stochastic generalization of enzyme inhibition kinetics to mesoscopic enzyme concentrations by systematically accounting for molecular noise in competitive and uncompetitive mechanisms of enzyme inhibition. Our work reveals an activator-inhibitor duality as a non-classical effect in the transient regime in which inhibitors tend to enhance enzymatic activity. We introduce statistical measures that quantify this counterintuitive response through the stochastic analog of the Lineweaver-Burk plot that shows a merging of the inhibitor-dependent velocity with the Michaelis-Menten velocity. The statistical measures of mean and temporal fluctuations - fractional enzyme activity and waiting time correlations - show a non-monotonic rise with the increase in inhibitors before subsiding to their baseline value. The inhibitor and substrate dependence of the fractional enzyme activity yields kinetic phase diagrams for non-classical activator-inhibitor duality. Our work links this duality to a molecular memory effect in the transient regime, arising from positive correlations between consecutive product turnover times. The vanishing of memory in the steady state recovers all the classical results.

Thermodynamic bounds on ultrasensitivity in covalent switching

Switch-like motifs are among the basic building blocks of biochemical networks. A common motif that can serve as an ultrasensitive switch consists of two enzymes acting antagonistically on a substrate, one making and the other removing a covalent modification. To work as a switch, such covalent modification cycles must be held out of thermodynamic equilibrium by continuous expenditure of energy. Here, we exploit the linear framework for timescale separation to establish tight bounds on the performance of any covalent-modification switch in terms of the chemical potential difference driving the cycle. The bounds apply to arbitrary enzyme mechanisms, not just Michaelis-Menten, with arbitrary rate constants and thereby reflect fundamental physical constraints on covalent switching.

Size limits the sensitivity of kinetic schemes

Living things benefit from exquisite molecular sensitivity in many of their key processes, including DNA replication, transcription and translation, chemical sensing, and morphogenesis. At thermodynamic equilibrium, the basic biophysical mechanism for sensitivity is cooperative binding, for which it can be shown that the Hill coefficient, a sensitivity measure, cannot exceed the number of binding sites. Generalizing this fact, we find that for any kinetic scheme, at or away from thermodynamic equilibrium, a very simple structural quantity, the size of the support of a perturbation, always limits the effective Hill coefficient. We show how this bound sheds light on and unifies diverse sensitivity mechanisms, including kinetic proofreading and a nonequilibrium Monod-Wyman-Changeux (MWC) model proposed for the E. coli flagellar motor switch, representing in each case a simple, precise bridge between experimental observations and the models we write down. In pursuit of mechanisms that saturate the support bound, we find a nonequilibrium binding mechanism, nested hysteresis, with sensitivity exponential in the number of binding sites, with implications for our understanding of models of gene regulation and the function of biomolecular condensates.

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.

An optimal growth law for RNA composition and its partial implementation through ribosomal and tRNA gene locations in bacterial genomes

The distribution of cellular resources across bacterial proteins has been quantified through phenomenological growth laws. Here, we describe a complementary bacterial growth law for RNA composition, emerging from optimal cellular resource allocation into ribosomes and ternary complexes. The predicted decline of the tRNA/rRNA ratio with growth rate agrees quantitatively with experimental data. Its regulation appears to be implemented in part through chromosomal localization, as rRNA genes are typically closer to the origin of replication than tRNA genes and thus have increasingly higher gene dosage at faster growth. At the highest growth rates in E. coli, the tRNA/rRNA gene dosage ratio based on chromosomal positions is almost identical to the observed and theoretically optimal tRNA/rRNA expression ratio, indicating that the chromosomal arrangement has evolved to favor maximal transcription of both types of genes at this condition.

Unlike the proteome composition, RNA composition is often assumed to be independent of growth rate in bacteria, despite experimental evidence for a growth rate dependence in many microbes. In this work, we derived a growth-rate dependent optimal tRNA/rRNA concentration ratio by minimizing the combined costs of ribosome and ternary complex at the required protein production rate. The predicted optimal tRNA/rRNA expression ratio, which is a monotonically decreasing function of growth rate, agrees with experimental data for E. coli and other fast-growing microbes. This indicates the existing of an RNA composition growth law. Due to the presence of partially replicated chromosomes, gene dosage is higher for those genes whose DNA is replicated earlier, an effect that becomes stronger at higher growth rates. Because rRNA genes are located closer to origin of replication than tRNA genes in fast-growing species, the tRNA/rRNA gene dosage ratio scales with growth rate in the same direction as the optimal tRNA/rRNA expression ratio. Thus, it appears that the RNA growth law is–at least in part–implemented simply through the genomic positions of tRNA and rRNA genes. This finding indicates that growth rate-dependent optimal resource allocation can influence the genomic organization in bacteria.

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.

Resetting transition is governed by an interplay between thermal and potential energy

A dynamical process that takes a random time to complete, e.g., a chemical reaction, may either be accelerated or hindered due to resetting. Tuning system parameters, such as temperature, viscosity, or concentration, can invert the effect of resetting on the mean completion time of the process, which leads to a resetting transition. Although the resetting transition has been recently studied for diffusion in a handful of model potentials, it is yet unknown whether the results follow any universality in terms of well-defined physical parameters. To bridge this gap, we propose a general framework that reveals that the resetting transition is governed by an interplay between the thermal and potential energy. This result is illustrated for different classes of potentials that are used to model a wide variety of stochastic processes with numerous applications.

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.

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.

A Cofactor-Substrate-Based Supramolecular Fluorescent Probe for the Ultrafast Detection of Nitroreductase under Hypoxic Conditions

Identifying the location and expression levels of enzymes under hypoxic conditions in cancer cells is vital in early-stage cancer diagnosis and monitoring. By encapsulating a fluorescent substrate, L-NO₂ , within the NADH mimic-containing metal-organic capsule Zn-MPB, we developed a cofactor-substrate-based supramolecular luminescent probe for ultrafast detection of hypoxia-related enzymes in solution in vitro and in vivo. The host-guest structure fuses the coenzyme and substrate into one supramolecular probe to avoid control by NADH, switching the catalytic process of nitroreductase from a double-substrate mechanism to a single-substrate one. This probe promotes enzyme efficiency by altering the substrate catalytic process and enhances the electron transfer efficiency through an intra-molecular pathway with increased activity. The enzyme content and fluorescence intensity showed a linear relationship and equilibrium was obtained in seconds, showing potential for early tumor diagnosis, biomimetic catalysis, and prodrug activation.

Bayesian inference and comparison of stochastic transcription elongation models

Transcription elongation can be modelled as a three step process, involving polymerase translocation, NTP binding, and nucleotide incorporation into the nascent mRNA. This cycle of events can be simulated at the single-molecule level as a continuous-time Markov process using parameters derived from single-molecule experiments. Previously developed models differ in the way they are parameterised, and in their incorporation of partial equilibrium approximations. We have formulated a hierarchical network comprised of 12 sequence-dependent transcription elongation models. The simplest model has two parameters and assumes that both translocation and NTP binding can be modelled as equilibrium processes. The most complex model has six parameters makes no partial equilibrium assumptions. We systematically compared the ability of these models to explain published force-velocity data, using approximate Bayesian computation. This analysis was performed using data for the RNA polymerase complexes of E. coli, S. cerevisiae and Bacteriophage T7. Our analysis indicates that the polymerases differ significantly in their translocation rates, with the rates in T7 pol being fast compared to E. coli RNAP and S. cerevisiae pol II. Different models are applicable in different cases. We also show that all three RNA polymerases have an energetic preference for the posttranslocated state over the pretranslocated state. A Bayesian inference and model selection framework, like the one presented in this publication, should be routinely applicable to the interrogation of single-molecule datasets.

Transcription is a critical biological process which occurs in all living organisms. It involves copying the organism’s genetic material into messenger RNA (mRNA) which directs protein synthesis on the ribosome. Transcription is performed by RNA polymerases which have been extensively studied using both ensemble and single-molecule techniques. Single-molecule data provides unique insights into the molecular behaviour of RNA polymerases. Transcription at the single-molecule level can be computationally simulated as a continuous-time Markov process and the model outputs compared with experimental data. In this study we use Bayesian techniques to perform a systematic comparison of 12 stochastic models of transcriptional elongation. We demonstrate how equilibrium approximations can strengthen or weaken the model, and show how Bayesian techniques can identify necessary or unnecessary model parameters. We describe a framework to a) simulate, b) perform inference on, and c) compare models of transcription elongation.

Negative reciprocity, not ordered assembly, underlies the interaction of Sox2 and Oct4 on DNA

The mode of interaction of transcription factors (TFs) on eukaryotic genomes remains a matter of debate. Single-molecule data in living cells for the TFs Sox2 and Oct4 were previously interpreted as evidence of ordered assembly on DNA. However, the quantity that was calculated does not determine binding order but, rather, energy expenditure away from thermodynamic equilibrium. Here, we undertake a rigorous biophysical analysis which leads to the concept of reciprocity. The single-molecule data imply that Sox2 and Oct4 exhibit negative reciprocity, with expression of Sox2 increasing Oct4’s genomic binding but expression of Oct4 decreasing Sox2’s binding. Models show that negative reciprocity can arise either from energy expenditure or from a mixture of positive and negative cooperativity at distinct genomic loci. Both possibilities imply unexpected complexity in how TFs interact on DNA, for which single-molecule methods provide novel detection capabilities.

The bodies of humans and other animals contain many types of cells that perform different roles in the body. Most cells in the body carry the same DNA, which is arranged into sections known as genes. The marked differences between cell types arise because different sets of genes are switched on or ‘expressed’.

Proteins called transcription factors control which genes are expressed by binding to DNA and recruiting groups of accessory proteins. However, it is not clear how they interact with each other and with the accessory proteins to decide whether to express a gene. For instance, it is thought that some accessory proteins may provide energy for this process, but it is unknown whether the energy is used continuously or only for a short time. Insights from physics suggest that the former could have more powerful effects.

In 2014, a team of researchers reported using a microscopy approach, known as single-molecule imaging, to follow two transcription factors called Sox2 and Oct4 in cells from mice. After analyzing the data, the researchers concluded that Sox2 and Oct4 had a specific order of binding to DNA, with Sox2 often binding first and then assisting Oct4 to bind. Now Biddle et al. report that the claim made in the 2014 study is unsupported because of errors in the way the data were analyzed. In particular, Biddle et al. argue that what the earlier study actually calculated is not an order of binding but a measure of whether energy is being continuously used when Sox2 and Oct4 bind to DNA.

Biddle et al. reanalyzed the data from the 2014 work and concluded that Sox2 increases the extent of Oct4 binding to DNA, while Oct4 decreases the amount of Sox2 binding to DNA. Mathematical models suggest this may be due to the continuous use of energy as the two proteins bind to DNA. Alternatively, it could also happen if Sox2 and Oct4 helped each other to bind at some sites on DNA and hindered each other from binding in other places, even if energy is only used for a short time. These findings reveal that there is unexpected complexity in how transcription factors bind to DNA.

The next step following on from this work is to carry out experiments that test the two possible explanations for how Sox2 and Oct4 interact on DNA. Including physics in the analysis may help describe more accurately the biology of how transcription factors determine gene expression. Understanding this process will shed new light on many important biological questions and may aid the development of gene therapy and other new medical techniques.