Protein logic: a statistical mechanical study of signal integration at the single-molecule level

Information processing and decision making is based upon logic operations, which in cellular networks has been well characterized at the level of transcription. In recent years however, both experimentalists and theorists have begun to appreciate that cellular decision making can also be performed at the level of a single protein, giving rise to the notion of protein logic. Here we systematically explore protein logic using a well known statistical mechanical model. As an example system, we focus on receptors which bind either one or two ligands, and their associated dimers. Notably, we find that a single heterodimer can realize any of the 16 possible logic gates, including the XOR gate, by variation of biochemical parameters. We then introduce the novel idea that a set of receptors with fixed parameters can encode functionally unique logic gates simply by forming different dimeric combinations. An exhaustive search reveals that the simplest set of receptors (two single-ligand receptors and one double-ligand receptor) can realize several different groups of three unique gates, a result for which the parametric analysis of single receptors and dimers provides a clear interpretation. Both results underscore the surprising functional freedom readily available to cells at the single-protein level.Comment: 19 pages, 4 figures and 9 pages S

Catalysis by hen egg-white lysozyme proceeds via a covalent intermediate

Hen egg-white lysozyme (HEWL) was the first enzyme to have its three-dimensional structure determined by X-ray diffraction techniques(1). A catalytic mechanism, featuring a long-lived oxo-carbenium-ion intermediate, was proposed on the basis of model-building studies(2). The `Phillips' mechanism is widely held as the paradigm for the catalytic mechanism of beta -glycosidases that cleave glycosidic linkages with net retention of configuration of the anomeric centre. Studies with other retaining beta -glycosidases, however, provide strong evidence pointing to a common mechanism for these enzymes that involves a covalent glycosyl-enzyme intermediate, as previously postulated(3). Here we show, in three different cases using electrospray ionization mass spectrometry, a catalytically competent covalent glycosyl-enzyme intermediate during the catalytic cycle of HEWL. We also show the three-dimensional structure of this intermediate as determined by Xray diffraction. We formulate a general catalytic mechanism for all retaining beta -glycosidases that includes substrate distortion, formation of a covalent intermediate, and the electrophilic migration of C1 along the reaction coordinate

Rigidity and flexibility of biological networks

The network approach became a widely used tool to understand the behaviour of complex systems in the last decade. We start from a short description of structural rigidity theory. A detailed account on the combinatorial rigidity analysis of protein structures, as well as local flexibility measures of proteins and their applications in explaining allostery and thermostability is given. We also briefly discuss the network aspects of cytoskeletal tensegrity. Finally, we show the importance of the balance between functional flexibility and rigidity in protein-protein interaction, metabolic, gene regulatory and neuronal networks. Our summary raises the possibility that the concepts of flexibility and rigidity can be generalized to all networks.Comment: 21 pages, 4 figures, 1 tabl

Safe uses of Hill's model: an exact comparison with the Adair-Klotz model

<p>Abstract</p> <p>Background</p> <p>The Hill function and the related Hill model are used frequently to study processes in the living cell. There are very few studies investigating the situations in which the model can be safely used. For example, it has been shown, at the mean field level, that the dose response curve obtained from a Hill model agrees well with the dose response curves obtained from a more complicated Adair-Klotz model, provided that the parameters of the Adair-Klotz model describe strongly cooperative binding. However, it has not been established whether such findings can be extended to other properties and non-mean field (stochastic) versions of the same, or other, models.</p> <p>Results</p> <p>In this work a rather generic quantitative framework for approaching such a problem is suggested. The main idea is to focus on comparing the particle number distribution functions for Hill's and Adair-Klotz's models instead of investigating a particular property (e.g. the dose response curve). The approach is valid for any model that can be mathematically related to the Hill model. The Adair-Klotz model is used to illustrate the technique. One main and two auxiliary similarity measures were introduced to compare the distributions in a quantitative way. Both time dependent and the equilibrium properties of the similarity measures were studied.</p> <p>Conclusions</p> <p>A strongly cooperative Adair-Klotz model can be replaced by a suitable Hill model in such a way that any property computed from the two models, even the one describing stochastic features, is approximately the same. The quantitative analysis showed that boundaries of the regions in the parameter space where the models behave in the same way exhibit a rather rich structure.</p

Cooperative Binding

Molecular binding is an interaction between molecules that results in a stable association between those molecules. Cooperative binding occurs if the number of binding sites of a macromolecule that are occupied by a specific type of ligand is a nonlinear function of this ligand’s concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supralinear) or negative (infralinear). Cooperative binding is most often observed in proteins, but nucleic acids can also exhibit cooperative binding, for instance of transcription factors. Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes

On the conservation of the slow conformational dynamics within the amino acid kinase family: NAGK the paradigm

N-Acetyl-L-Glutamate Kinase (NAGK) is the structural paradigm for examining the catalytic mechanisms and dynamics of amino acid kinase family members. Given that the slow conformational dynamics of the NAGK (at the microseconds time scale or slower) may be rate-limiting, it is of importance to assess the mechanisms of the most cooperative modes of motion intrinsically accessible to this enzyme. Here, we present the results from normal mode analysis using an elastic network model representation, which shows that the conformational mechanisms for substrate binding by NAGK strongly correlate with the intrinsic dynamics of the enzyme in the unbound form. We further analyzed the potential mechanisms of allosteric signalling within NAGK using a Markov model for network communication. Comparative analysis of the dynamics of family members strongly suggests that the low-frequency modes of motion and the associated intramolecular couplings that establish signal transduction are highly conserved among family members, in support of the paradigm sequence→structure→dynamics→function © 2010 Marcos et al

Chemotactic response and adaptation dynamics in Escherichia coli

Adaptation of the chemotaxis sensory pathway of the bacterium Escherichia coli is integral for detecting chemicals over a wide range of background concentrations, ultimately allowing cells to swim towards sources of attractant and away from repellents. Its biochemical mechanism based on methylation and demethylation of chemoreceptors has long been known. Despite the importance of adaptation for cell memory and behavior, the dynamics of adaptation are difficult to reconcile with current models of precise adaptation. Here, we follow time courses of signaling in response to concentration step changes of attractant using in vivo fluorescence resonance energy transfer measurements. Specifically, we use a condensed representation of adaptation time courses for efficient evaluation of different adaptation models. To quantitatively explain the data, we finally develop a dynamic model for signaling and adaptation based on the attractant flow in the experiment, signaling by cooperative receptor complexes, and multiple layers of feedback regulation for adaptation. We experimentally confirm the predicted effects of changing the enzyme-expression level and bypassing the negative feedback for demethylation. Our data analysis suggests significant imprecision in adaptation for large additions. Furthermore, our model predicts highly regulated, ultrafast adaptation in response to removal of attractant, which may be useful for fast reorientation of the cell and noise reduction in adaptation.Comment: accepted for publication in PLoS Computational Biology; manuscript (19 pages, 5 figures) and supplementary information; added additional clarification on alternative adaptation models in supplementary informatio

A danger of low copy numbers for inferring incorrect cooperativity degree

Background: A dose-response curve depicts fraction of bound proteins as a function of unbound ligands. Dose-response curves are used to measure the cooperativity degree of a ligand binding process. Frequently, the Hill function is used to fit the experimental data. The Hill function is parameterized by the value of the dissociation constant, and the Hill coefficient which describes the cooperativity degree. The use of Hill's model and the Hill function have been heavily criticised in this context, predominantly the assumption that all ligands bind at once, which lead to further refinements of the model. In this work, the validity of the Hill function has been studied from an entirely different point of view. In the limit of low copy numbers the dynamics of the system becomes noisy. The goal was to asses the validity of the Hill function in this limit, and to see in which ways the effects of the fluctuations change the form of the dose-response curves. Results: Dose-response curves were computed taking into account effects of fluctuations. The effects of fluctuations were described at the lowest order (the second moment of the particle number distribution) by using previously developed Pair Approach Reaction Noise EStimator (PARNES) method. The stationary state of the system is described by nine equations with nine unknowns. To obtain fluctuation corrected dose-response curves the equations have been investigated numerically. Conclusions: The Hill function cannot describe dose-response curves in a low particle limit. First, dose-response curves are not solely parameterized by the dissociation constant and the Hill coefficient. In general, the shape of a dose-response curve depends on the variables that describe how an experiment (ensemble) is designed. Second, dose-response curves are multi valued in a rather non-trivial way

Saturation Behavior: a general relationship described by a simple second-order differential equation

<p>Abstract</p> <p>Background</p> <p>The numerous natural phenomena that exhibit saturation behavior, <it>e.g</it>., ligand binding and enzyme kinetics, have been approached, to date, via empirical and particular analyses. This paper presents a mechanism-free, and assumption-free, second-order differential equation, designed only to describe a typical relationship between the variables governing these phenomena. It develops a mathematical model for this relation, based solely on the analysis of the typical experimental data plot and its saturation characteristics. Its utility complements the traditional empirical approaches.</p> <p>Results</p> <p>For the general saturation curve, described in terms of its independent (<it>x</it>) and dependent (<it>y</it>) variables, a second-order differential equation is obtained that applies to any saturation phenomena. It shows that the driving factor for the basic saturation behavior is the probability of the interactive site being free, which is described quantitatively. Solving the equation relates the variables in terms of the two empirical constants common to all these phenomena, the initial slope of the data plot and the limiting value at saturation. A first-order differential equation for the slope emerged that led to the concept of the effective binding rate at the active site and its dependence on the calculable probability the interactive site is free. These results are illustrated using specific cases, including ligand binding and enzyme kinetics. This leads to a revised understanding of how to interpret the empirical constants, in terms of the variables pertinent to the phenomenon under study.</p> <p>Conclusions</p> <p>The second-order differential equation revealed the basic underlying relations that describe these saturation phenomena, and the basic mathematical properties of the standard experimental data plot. It was shown how to integrate this differential equation, and define the common basic properties of these phenomena. The results regarding the importance of the slope and the new perspectives on the empirical constants governing the behavior of these phenomena led to an alternative perspective on saturation behavior kinetics. Their essential commonality was revealed by this analysis, based on the second-order differential equation.</p

Preference of Small Molecules for Local Minimum Conformations when Binding to Proteins

It is well known that small molecules (ligands) do not necessarily adopt their lowest potential energy conformations when binding to proteins. Analyses of protein-bound ligand crystal structures have reportedly shown that many of them do not even adopt the conformations at local minima of their potential energy surfaces (local minimum conformations). The results of these analyses raise a concern regarding the validity of virtual screening methods that use ligands in local minimum conformations. Here we report a normal-mode-analysis (NMA) study of 100 crystal structures of protein-bound ligands. Our data show that the energy minimization of a ligand alone does not automatically stop at a local minimum conformation if the minimum of the potential energy surface is shallow, thus leading to the folding of the ligand. Furthermore, our data show that all 100 ligand conformations in their protein-bound ligand crystal structures are nearly identical to their local minimum conformations obtained from NMA-monitored energy minimization, suggesting that ligands prefer to adopt local minimum conformations when binding to proteins. These results both support virtual screening methods that use ligands in local minimum conformations and caution about possible adverse effect of excessive energy minimization when generating a database of ligand conformations for virtual screening