Some lessons about models from Michaelis and Menten

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.

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.

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.

A guide to the Michaelis-Menten equation: steady state and beyond

The modern definition of enzymology is synonymous with the Michaelis-Menten equation instituted by Leonor Michaelis and Maud Menten. Most textbooks, or chapters within, discussing enzymology start with the derivation of the equation under the assumption of rapid equilibrium (as done by Michaelis-Menten) or steady state (as modified by Briggs and Haldane) conditions to highlight the importance of this equation as the bedrock on which interpretation of enzyme kinetic results is dependent. However, few textbooks or monographs take the effort of placing the equation within its right historical context and discuss the assumptions that have gone into its institution. This guide will dwell on these in substantial detail. Further, this guide will attempt to instil a sense of appreciation for the mathematical curve rectangular hyperbola, its unique attributes and how ubiquitous the curve is in biological systems. To conclude, this guide will discuss the limitations of the equation, and the method it embodies, and trace the journey of how investigators are attempting to move beyond the steady-state approach and the Michaelis-Menten equation into full progress curve, pre-steady state and single-turnover kinetic analysis to obtain greater insights into enzyme kinetics and catalysis.

A forecast for large-scale, predictive biology: Lessons from meteorology

Quantitative systems biology, in which predictive mathematical models are constructed to guide the design of experiments and predict experimental outcomes, is at an exciting transition point, where the foundational scientific principles are becoming established, but the impact is not yet global. The next steps necessary for mathematical modeling to transform biological research and applications, in the same way it has already transformed other fields, is not completely clear. The purpose of this perspective is to forecast possible answers to this question-what needs to happen next-by drawing on the experience gained in another field, specifically meteorology. We review here a number of lessons learned in weather prediction that are directly relevant to biological systems modeling, and that we believe can enable the same kinds of global impact in our field as atmospheric modeling makes today.

Modeling Resilience and Sustainability of Water-Subsidized Systems: An Example from Northwest Costa Rica

Water-subsidized systems are growing in number and maintaining the sustainability of such complex systems presents unique challenges. Interbasin water transfer creates new sociohydrological dynamics that come with tradeoffs and potential regime shifts. The Tempisque-Bebedero watershed in Northwest Costa Rica typifies this class of watershed: Transferred water is used for power generation and irrigated agriculture with significant downstream environmental impacts. To improve and clarify our understanding of the effects of social and biophysical factors on the resilience of such systems, a stylized dynamical systems model was developed, using as a guide the situation in the Tempisque-Bebedero watershed. This model was analyzed to understand the nature of socio-hydrologic regimes that exist in this class of basins and what factors determine these regimes. The model analysis revealed five distinct regimes and different regime shift behaviors dependent on environmental and policy conditions. This work offers insights into other complex socio-hydrologic systems with similar processes.

Major Improvements in Robustness and Efficiency during the Screening of Novel Enzyme Effectors by the 3-Point Kinetics Assay

The throughput level currently reached by automatic liquid handling and assay monitoring techniques is expected to facilitate the discovery of new modulators of enzyme activity. Judicious and dependable ways to interpret vast amounts of information are, however, required to effectively answer this challenge. Here, the 3-point method of kinetic analysis is proposed as a means to significantly increase the hit success rates and decrease the number of falsely identified compounds (false positives). In this post-Michaelis-Menten approach, each screened reaction is probed in three different occasions, none of which necessarily coincide with the initial period of constant velocity. Enzymology principles rather than subjective criteria are applied to identify unwanted outliers such as assay artifacts, and then to accurately distinguish true enzyme modulation effects from false positives. The exclusion and selection criteria are defined based on the 3-point reaction coordinates, whose relative positions along the time-courses may change from well to well or from plate to plate, if necessary. The robustness and efficiency of the new method is illustrated during a small drug repurposing screening of potential modulators of the deubiquinating activity of ataxin-3, a protein implicated in Machado-Joseph disease. Apparently, intractable Z factors are drastically enhanced after (1) eliminating spurious results, (2) improving the normalization method, and (3) increasing the assay resilience to systematic and random variability. Numerical simulations further demonstrate that the 3-point analysis is highly sensitive to specific, catalytic, and slow-onset modulation effects that are particularly difficult to detect by typical endpoint assays.

Effect of UVB solar irradiation on Laccase enzyme: evaluation of the photooxidation process and its impact over the enzymatic activity for pollutants bioremediation

The multi-copper Laccase enzyme corresponds to one of the most investigated oxidoreductases for potential uses in xenobiotic bioremediation. In this work, we have investigated the photo-degradation process of Laccase from Trametesversicolor induced by UVB light and the influence on its activity over selected substrates. Laccase undergoes photo-degradation when irradiated with UVB light, and the process depends on the presence of oxygen in the medium. With the kinetic data obtained from stationary and time resolved measurements, a photo-degradation mechanism of auto-sensitization was proposed for the enzyme. Laccase generates singlet oxygen, by UVB light absorption, and this reactive oxygen species can trigger the photo-oxidation of susceptible amino acids residues present in the protein structure. The catalytic activity of Laccase was evaluated before and after UVB photolysis over hydroxy-aromatic compounds and substituted phenols which represent potential pollutants. The dye bromothymol blue, the antibiotic rifampicin and the model compound syringaldazine, were selected as substrates. The values of the kinetic parameters determined in our experiments indicate that the photo-oxidative process of Laccase has a very negative impact on its overall catalytic function. Despite this, we have not found evidence of structural damage by SDS-PAGE and circular dichroism experiments, which indicate that the enzyme retained its secondary structure. We believe that, given the importance of Laccase in environmental bioremediation, the information found about the stability of this kind of biomolecule exposed to UV solar irradiation may be relevant in the technological design and/or optimization of decontamination strategies.

Robustness and parameter geography in post-translational modification systems

Biological systems are acknowledged to be robust to perturbations but a rigorous understanding of this has been elusive. In a mathematical model, perturbations often exert their effect through parameters, so sizes and shapes of parametric regions offer an integrated global estimate of robustness. Here, we explore this “parameter geography” for bistability in post-translational modification (PTM) systems. We use the previously developed “linear framework” for timescale separation to describe the steady-states of a two-site PTM system as the solutions of two polynomial equations in two variables, with eight non-dimensional parameters. Importantly, this approach allows us to accommodate enzyme mechanisms of arbitrary complexity beyond the conventional Michaelis-Menten scheme, which unrealistically forbids product rebinding. We further use the numerical algebraic geometry tools Bertini, Paramotopy, and alphaCertified to statistically assess the solutions to these equations at ∼10⁹ parameter points in total. Subject to sampling limitations, we find no bistability when substrate amount is below a threshold relative to enzyme amounts. As substrate increases, the bistable region acquires 8-dimensional volume which increases in an apparently monotonic and sigmoidal manner towards saturation. The region remains connected but not convex, albeit with a high visibility ratio. Surprisingly, the saturating bistable region occupies a much smaller proportion of the sampling domain under mechanistic assumptions more realistic than the Michaelis-Menten scheme. We find that bistability is compromised by product rebinding and that unrealistic assumptions on enzyme mechanisms have obscured its parametric rarity. The apparent monotonic increase in volume of the bistable region remains perplexing because the region itself does not grow monotonically: parameter points can move back and forth between monostability and bistability. We suggest mathematical conjectures and questions arising from these findings. Advances in theory and software now permit insights into parameter geography to be uncovered by high-dimensional, data-centric analysis.

Biological organisms are often said to have robust properties but it is difficult to understand how such robustness arises from molecular interactions. Here, we use a mathematical model to study how the molecular mechanism of protein modification exhibits the property of multiple internal states, which has been suggested to underlie memory and decision making. The robustness of this property is revealed by the size and shape, or “geography,” of the parametric region in which the property holds. We use advances in reducing model complexity and in rapidly solving the underlying equations, to extensively sample parameter points in an 8-dimensional space. We find that under realistic molecular assumptions the size of the region is surprisingly small, suggesting that generating multiple internal states with such a mechanism is much harder than expected. While the shape of the region appears straightforward, we find surprising complexity in how the region grows with increasing amounts of the modified substrate. Our approach uses statistical analysis of data generated from a model, rather than from experiments, but leads to precise mathematical conjectures about parameter geography and biological robustness.

An efficient wavelet-based optimization algorithm for the solutions of reaction-diffusion equations in biomedicine

In this paper, a mathematical model of nonlinear reaction-diffusion equation with Michaelis-Menten kinetics in a solid of planar and spherical shape is discussed. The proposed model is based on non-stationary diffusion equation containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reaction. An efficient wavelet-based spectral method has been developed for the analytical expressions pertaining to substrate concentration for all parameter values. The efficiency of the proposed wavelet method is confirmed by mean of the computational CPU time. The proposed wavelet-based results are compared with Adomian Decomposition Method (ADM). Satisfactory agreement with ADM results is observed. Moreover, the use of the wavelet method is found to be simple, efficient, flexible, and straight forward. Also, it requires less computation costs.

From Gregor Mendel to Eric Davidson: Mathematical Models and Basic Principles in Biology

Mathematical models have been widespread in biology since its emergence as a modern experimental science in the 19th century. Focusing on models in developmental biology and heredity, this article (1) presents the properties and epistemological basis of pertinent mathematical models in biology from Mendel's model of heredity in the 19th century to Eric Davidson's model of developmental gene regulatory networks in the 21st; (2) shows that the models differ not only in their epistemologies but also in regard to explicitly or implicitly taking into account basic biological principles, in particular those of biological specificity (that became, in part, replaced by genetic information) and genetic causality. The article claims that models disregarding these principles did not impact the direction of biological research in a lasting way, although some of them, such as D'Arcy Thompson's models of biological form, were widely read and admired and others, such as Turing's models of development, stimulated research in other fields. Moreover, it suggests that successful models were not purely mathematical descriptions or simulations of biological phenomena but were based on inductive, as well as hypothetico-deductive, methodology. The recent availability of large amounts of sequencing data and new computational methodology tremendously facilitates system approaches and pattern recognition in many fields of research. Although these new technologies have given rise to claims that correlation is replacing experimentation and causal analysis, the article argues that the inductive and hypothetico-deductive experimental methodologies have remained fundamentally important as long as causal-mechanistic explanations of complex systems are pursued.

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Integrative biological simulation praxis: Considerations from physics, philosophy, and data/model curation practices

Integrative biological simulations have a varied and controversial history in the biological sciences. From computational models of organelles, cells, and simple organisms, to physiological models of tissues, organ systems, and ecosystems, a diverse array of biological systems have been the target of large-scale computational modeling efforts. Nonetheless, these research agendas have yet to prove decisively their value among the broader community of theoretical and experimental biologists. In this commentary, we examine a range of philosophical and practical issues relevant to understanding the potential of integrative simulations. We discuss the role of theory and modeling in different areas of physics and suggest that certain sub-disciplines of physics provide useful cultural analogies for imagining the future role of simulations in biological research. We examine philosophical issues related to modeling which consistently arise in discussions about integrative simulations and suggest a pragmatic viewpoint that balances a belief in philosophy with the recognition of the relative infancy of our state of philosophical understanding. Finally, we discuss community workflow and publication practices to allow research to be readily discoverable and amenable to incorporation into simulations. We argue that there are aligned incentives in widespread adoption of practices which will both advance the needs of integrative simulation efforts as well as other contemporary trends in the biological sciences, ranging from open science and data sharing to improving reproducibility.

The (Mathematical) Modeling Process in Biosciences

In this communication, we introduce a general framework and discussion on the role of models and the modeling process in the field of biosciences. The objective is to sum up the common procedures during the formalization and analysis of a biological problem from the perspective of Systems Biology, which approaches the study of biological systems as a whole. We begin by presenting the definitions of (biological) system and model. Particular attention is given to the meaning of mathematical model within the context of biology. Then, we present the process of modeling and analysis of biological systems. Three stages are described in detail: conceptualization of the biological system into a model, mathematical formalization of the previous conceptual model and optimization and system management derived from the analysis of the mathematical model. All along this work the main features and shortcomings of the process are analyzed and a set of rules that could help in the task of modeling any biological system are presented. Special regard is given to the formative requirements and the interdisciplinary nature of this approach. We conclude with some general considerations on the challenges that modeling is posing to current biology.

Beware the tail that wags the dog: informal and formal models in biology

Informal models have always been used in biology to guide thinking and devise experiments. In recent years, formal mathematical models have also been widely introduced. It is sometimes suggested that formal models are inherently superior to informal ones and that biology should develop along the lines of physics or economics by replacing the latter with the former. Here I suggest to the contrary that progress in biology requires a better integration of the formal with the informal.

Invariants reveal multiple forms of robustness in bifunctional enzyme systems

Experimental and theoretical studies have suggested that bifunctional enzymes catalyzing opposing modification and demodification reactions can confer steady-state concentration robustness to their substrates. However, the types of robustness and the biochemical basis for them have remained elusive. Here we report a systematic study of the most general biochemical reaction network for a bifunctional enzyme acting on a substrate with one modification site, along with eleven sub-networks with more specialized biochemical assumptions. We exploit ideas from computational algebraic geometry, introduced in previous work, to find a polynomial expression (an invariant) between the steady state concentrations of the modified and unmodified substrate for each network. We use these invariants to identify five classes of robust behavior: robust upper bounds on concentration, robust two-sided bounds on concentration ratio, hybrid robustness, absolute concentration robustness (ACR), and robust concentration ratio. This analysis demonstrates that robustness can take a variety of forms and that the type of robustness is sensitive to many biochemical details, with small changes in biochemistry leading to very different steady-state behaviors. In particular, we find that the widely-studied ACR requires highly specialized assumptions in addition to bifunctionality. An unexpected result is that the robust bounds derived from invariants are strictly tighter than those derived by ad hoc manipulation of the underlying differential equations, confirming the value of invariants as a tool to gain insight into biochemical reaction networks. Furthermore, invariants yield multiple experimentally testable predictions and illuminate new strategies for inferring enzymatic mechanisms from steady-state measurements.

Stochastic simulation in systems biology

Natural systems are, almost by definition, heterogeneous: this can be either a boon or an obstacle to be overcome, depending on the situation. Traditionally, when constructing mathematical models of these systems, heterogeneity has typically been ignored, despite its critical role. However, in recent years, stochastic computational methods have become commonplace in science. They are able to appropriately account for heterogeneity; indeed, they are based around the premise that systems inherently contain at least one source of heterogeneity (namely, intrinsic heterogeneity). In this mini-review, we give a brief introduction to theoretical modelling and simulation in systems biology and discuss the three different sources of heterogeneity in natural systems. Our main topic is an overview of stochastic simulation methods in systems biology. There are many different types of stochastic methods. We focus on one group that has become especially popular in systems biology, biochemistry, chemistry and physics. These discrete-state stochastic methods do not follow individuals over time; rather they track only total populations. They also assume that the volume of interest is spatially homogeneous. We give an overview of these methods, with a discussion of the advantages and disadvantages of each, and suggest when each is more appropriate to use. We also include references to software implementations of them, so that beginners can quickly start using stochastic methods for practical problems of interest.

ERK as a model for systems biology of enzyme kinetics in cells

A key step towards a chemical picture of enzyme catalysis was taken in 1913, when Leonor Michaelis and Maud Menten published their studies of sucrose hydrolysis by invertase. Based on a novel experimental design and a mathematical model, their work offered a quantitative view of biochemical kinetics well before the protein nature of enzymes was established and complexes with substrates could be detected. Michaelis-Menten kinetics provides a solid framework for enzyme kinetics in vitro, but what about kinetics in cells, where enzymes can be highly regulated and participate in a multitude of interactions? We discuss this question using the Extracellular Signal Regulated Kinase (ERK), which controls a myriad functions in cells, as a model of an important enzyme for which we have crystal structures, quantitative in vitro assays, and a vast list of binding partners. Despite great progress, we still cannot quantitatively predict how the rates of ERK-dependent reactions respond to genetic and pharmacological perturbations. Achieving this goal, which is important from both fundamental and practical standpoints, requires measuring the rates of enzyme reactions in their native environment and interpreting these measurements using simple but realistic mathematical models--the two elements which served as the cornerstones for Michaelis' and Menten's seminal 1913 paper.

Models in biology: 'accurate descriptions of our pathetic thinking'

In this essay I will sketch some ideas for how to think about models in biology. I will begin by trying to dispel the myth that quantitative modeling is somehow foreign to biology. I will then point out the distinction between forward and reverse modeling and focus thereafter on the former. Instead of going into mathematical technicalities about different varieties of models, I will focus on their logical structure, in terms of assumptions and conclusions. A model is a logical machine for deducing the latter from the former. If the model is correct, then, if you believe its assumptions, you must, as a matter of logic, also believe its conclusions. This leads to consideration of the assumptions underlying models. If these are based on fundamental physical laws, then it may be reasonable to treat the model as 'predictive', in the sense that it is not subject to falsification and we can rely on its conclusions. However, at the molecular level, models are more often derived from phenomenology and guesswork. In this case, the model is a test of its assumptions and must be falsifiable. I will discuss three models from this perspective, each of which yields biological insights, and this will lead to some guidelines for prospective model builders.

Role of substrate unbinding in Michaelis-Menten enzymatic reactions

The Michaelis-Menten equation provides a hundred-year-old prediction by which any increase in the rate of substrate unbinding will decrease the rate of enzymatic turnover. Surprisingly, this prediction was never tested experimentally nor was it scrutinized using modern theoretical tools. Here we show that unbinding may also speed up enzymatic turnover--turning a spotlight to the fact that its actual role in enzymatic catalysis remains to be determined experimentally. Analytically constructing the unbinding phase space, we identify four distinct categories of unbinding: inhibitory, excitatory, superexcitatory, and restorative. A transition in which the effect of unbinding changes from inhibitory to excitatory as substrate concentrations increase, and an overlooked tradeoff between the speed and efficiency of enzymatic reactions, are naturally unveiled as a result. The theory presented herein motivates, and allows the interpretation of, groundbreaking experiments in which existing single-molecule manipulation techniques will be adapted for the purpose of measuring enzymatic turnover under a controlled variation of unbinding rates. As we hereby show, these experiments will not only shed first light on the role of unbinding but will also allow one to determine the time distribution required for the completion of the catalytic step in isolation from the rest of the enzymatic turnover cycle.

Biology is more theoretical than physics

The word "theory" is used in at least two senses--to denote a body of widely accepted laws or principles, as in "Darwinian theory" or "quantum theory," and to suggest a speculative hypothesis, often relying on mathematical analysis, that has not been experimentally confirmed. It is often said that there is no place for the second kind of theory in biology and that biology is not theoretical but based on interpretation of data. Here, ideas from a previous essay are expanded upon to suggest, to the contrary, that the second kind of theory has always played a critical role and that biology, therefore, is a good deal more theoretical than physics.

Time-scale separation--Michaelis and Menten's old idea, still bearing fruit

Michaelis and Menten introduced to biochemistry the idea of time-scale separation, in which part of a system is assumed to be operating sufficiently fast compared to the rest so that it may be taken to have reached a steady state. This allows, in principle, the fast components to be eliminated, resulting in a simplified description of the system's behaviour. Similar ideas have been widely used in different areas of biology, including enzyme kinetics, protein allostery, receptor pharmacology, gene regulation and post-translational modification. However, the methods used have been independent and ad hoc. In the present study, we review the use of time-scale separation as a means to simplify the description of molecular complexity and discuss recent work setting out a single framework that unifies these separate calculations. The framework offers new capabilities for mathematical analysis and helps to do justice to Michaelis and Menten's insights about individual enzymes in the context of multi-enzyme biological systems.

Challenges for an enzymatic reaction kinetics database

The scientific literature contains a tremendous amount of kinetic data describing the dynamic behaviour of biochemical reactions over time. These data are needed for computational modelling to create models of biochemical reaction networks and to obtain a better understanding of the processes in living cells. To extract the knowledge from the literature, biocurators are required to understand a paper and interpret the data. For modellers, as well as experimentalists, this process is very time consuming because the information is distributed across the publication and, in most cases, is insufficiently structured and often described without standard terminology. In recent years, biological databases for different data types have been developed. The advantages of these databases lie in their unified structure, searchability and the potential for augmented analysis by software, which supports the modelling process. We have developed the SABIO-RK database for biochemical reaction kinetics. In the present review, we describe the challenges for database developers and curators, beginning with an analysis of relevant publications up to the export of database information in a standardized format. The aim of the present review is to draw the experimentalist's attention to the problem (from a data integration point of view) of incompletely and imprecisely written publications. We describe how to lower the barrier to curators and improve this situation. At the same time, we are aware that curating experimental data takes time. There is a community concerned with making the task of publishing data with the proper structure and annotation to ontologies much easier. In this respect, we highlight some useful initiatives and tools.

Logic modeling and the ridiculome under the rug

Logic-derived modeling has been used to map biological networks and to study arbitrary functional interactions, and fine-grained kinetic modeling can accurately predict the detailed behavior of well-characterized molecular systems; at present, however, neither approach comes close to unraveling the full complexity of a cell. The current data revolution offers significant promises and challenges to both approaches - and could bring them together as it has spurred the development of new methods and tools that may help to bridge the many gaps between data, models, and mechanistic understanding.

Have you used logic modeling in your research? It would not be surprising if many biologists would answer no to this hypothetical question. And it would not be true. In high school biology we already became familiar with cartoon diagrams that illustrate basic mechanisms of the molecular machinery operating inside cells. These are nothing else but simple logic models. If receptor and ligand are present, then receptor-ligand complexes form; if a receptor-ligand complex exists, then an enzyme gets activated; if the enzyme is active, then a second messenger is being produced; and so on. Such chains of causality are the essence of logic models (Figure 1a). Arbitrary events and mechanisms are abstracted; relationships are simplified and usually involve just two possible conditions and three possible consequences. The presence or absence of one or more molecule, activity, or function, [some icons in the cartoon] will determine whether another one of them will be produced (created, up-regulated, stimulated) [a 'positive' link] or destroyed (degraded, down-regulated, inhibited) [a 'negative' link], or be unaffected [there is no link]. The icons and links often do not follow a standardized format, but when we look at such a cartoon diagram, we believe that we 'understand' how the system works. Because our brain is easily able to process these relationships, these diagrams allow us to answer two fundamental types of questions related to the system: why (are certain things happening)? What if (we make some changes)?

Optimization of enzyme parameters for fermentative production of biorenewable fuels and chemicals

Microbial biocatalysts such as Escherichia coli and Saccharomyces cerevisiae have been extensively subjected to Metabolic Engineering for the fermentative production of biorenewable fuels and chemicals. This often entails the introduction of new enzymes, deletion of unwanted enzymes and efforts to fine-tune enzyme abundance in order to attain the desired strain performance. Enzyme performance can be quantitatively described in terms of the Michaelis-Menten type parameters K_m, turnover number k_cat and K_i, which roughly describe the affinity of an enzyme for its substrate, the speed of a reaction and the enzyme sensitivity to inhibition by regulatory molecules. Here we describe examples of where knowledge of these parameters have been used to select, evolve or engineer enzymes for the desired performance and enabled increased production of biorenewable fuels and chemicals. Examples include production of ethanol, isobutanol, 1-butanol and tyrosine and furfural tolerance. The Michaelis-Menten parameters can also be used to judge the cofactor dependence of enzymes and quantify their preference for NADH or NADPH. Similarly, enzymes can be selected, evolved or engineered for the preferred cofactor preference. Examples of exporter engineering and selection are also discussed in the context of production of malate, valine and limonene.

Post-translational modification: nature's escape from genetic imprisonment and the basis for dynamic information encoding

We discuss protein post-translational modification (PTM) from an information processing perspective. PTM at multiple sites on a protein creates a combinatorial explosion in the number of potential 'mod-forms', or global patterns of modification. Distinct mod-forms can elicit distinct downstream responses, so that the overall response depends partly on the effectiveness of a particular mod-form to elicit a response and partly on the stoichiometry of that mod-form in the molecular population. We introduce the 'mod-form distribution'-the relative stoichiometries of each mod-form-as the most informative measure of a protein's state. Distinct mod-form distributions may summarize information about distinct cellular and physiological conditions and allow downstream processes to interpret this information accordingly. Such information 'encoding' by PTMs may facilitate evolution by weakening the need to directly link upstream conditions to downstream responses. Mod-form distributions provide a quantitative framework in which to interpret ideas of 'PTM codes' that are emerging in several areas of biology, as we show by reviewing examples of ion channels, GPCRs, microtubules, and transcriptional co-regulators. We focus particularly on examples other than the well-known 'histone code', to emphasize the pervasive use of information encoding in molecular biology. Finally, we touch briefly on new methods for measuring mod-form distributions.

Realistic enzymology for post-translational modification: zero-order ultrasensitivity revisited

Unlimited ultrasensitivity in a kinase/phosphatase "futile cycle" has been a paradigmatic example of collective behaviour in multi-enzyme systems. However, its analysis has relied on the Michaelis-Menten reaction mechanism, which remains widely used despite a century of new knowledge. Modifying and demodifying enzymes accomplish different biochemical tasks; the donor that contributes the modifying group is often ignored without the impact of this time-scale separation being taken into account; and new forms of reversible modification are now known. We exploit new algebraic methods of steady-state analysis to reconcile the analysis of multi-enzyme systems with single-enzyme biochemistry using zero-order ultrasensitivity as an example. We identify the property of "strong irreversibility", in which product re-binding is disallowed. We show that unlimited ultrasensitivity is preserved for a class of complex, strongly irreversible reaction mechanisms and determine the corresponding saturation conditions. We show further that unlimited ultrasensitivity arises from a singularity in a novel "invariant" that summarises the algebraic relationship between modified and unmodified substrate. We find that this singularity also underlies knife-edge behaviour in allocation of substrate between modification states, which has implications for the coherence of futile cycles within an integrated tissue. When the enzymes are irreversible, but not strongly so, the singularity disappears in the form found here and unlimited ultrasensitivity may no longer be preserved. The methods introduced here are widely applicable to other reversible modification systems.

A linear framework for time-scale separation in nonlinear biochemical systems

Cellular physiology is implemented by formidably complex biochemical systems with highly nonlinear dynamics, presenting a challenge for both experiment and theory. Time-scale separation has been one of the few theoretical methods for distilling general principles from such complexity. It has provided essential insights in areas such as enzyme kinetics, allosteric enzymes, G-protein coupled receptors, ion channels, gene regulation and post-translational modification. In each case, internal molecular complexity has been eliminated, leading to rational algebraic expressions among the remaining components. This has yielded familiar formulas such as those of Michaelis-Menten in enzyme kinetics, Monod-Wyman-Changeux in allostery and Ackers-Johnson-Shea in gene regulation. Here we show that these calculations are all instances of a single graph-theoretic framework. Despite the biochemical nonlinearity to which it is applied, this framework is entirely linear, yet requires no approximation. We show that elimination of internal complexity is feasible when the relevant graph is strongly connected. The framework provides a new methodology with the potential to subdue combinatorial explosion at the molecular level.