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

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

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 half-site multimeric enzyme achieves its cooperativity without conformational changes

Cooperativity is a feature many multimeric proteins use to control activity. Here we show that the bacterial heptose isomerase GmhA displays homotropic positive and negative cooperativity among its four protomers. Most similar proteins achieve this through conformational changes: GmhA instead employs a delicate network of hydrogen bonds, and couples pairs of active sites controlled by a unique water channel. This network apparently raises the Lewis acidity of the catalytic zinc, thus increasing the activity at one active site at the cost of preventing substrate from adopting a reactive conformation at the paired negatively cooperative site – a “half-site” behavior. Our study establishes the principle that multimeric enzymes can exploit this cooperativity without conformational changes to maximize their catalytic power and control. More broadly, this subtlety by which enzymes regulate functions could be used to explore new inhibitor design strategies

Trade-Offs and Constraints in Allosteric Sensing

Sensing extracellular changes initiates signal transduction and is the first stage of cellular decision-making. Yet relatively little is known about why one form of sensing biochemistry has been selected over another. To gain insight into this question, we studied the sensing characteristics of one of the biochemically simplest of sensors: the allosteric transcription factor. Such proteins, common in microbes, directly transduce the detection of a sensed molecule to changes in gene regulation. Using the Monod-Wyman-Changeux model, we determined six sensing characteristics – the dynamic range, the Hill number, the intrinsic noise, the information transfer capacity, the static gain, and the mean response time – as a function of the biochemical parameters of individual sensors and of the number of sensors. We found that specifying one characteristic strongly constrains others. For example, a high dynamic range implies a high Hill number and a high capacity, and vice versa. Perhaps surprisingly, these constraints are so strong that most of the space of characteristics is inaccessible given biophysically plausible ranges of parameter values. Within our approximations, we can calculate the probability distribution of the numbers of input molecules that maximizes information transfer and show that a population of one hundred allosteric transcription factors can in principle distinguish between more than four bands of input concentrations. Our results imply that allosteric sensors are unlikely to have been selected for high performance in one sensing characteristic but for a compromise in the performance of many

Specificity quantification of biomolecular recognition and its implication for drug discovery

Highly efficient and specific biomolecular recognition requires both affinity and specificity. Previous quantitative descriptions of biomolecular recognition were mostly driven by improving the affinity prediction, but lack of quantification of specificity. We developed a novel method SPA (SPecificity and Affinity) based on our funneled energy landscape theory. The strategy is to simultaneously optimize the quantified specificity of the “native” protein-ligand complex discriminating against “non-native” binding modes and the affinity prediction. The benchmark testing of SPA shows the best performance against 16 other popular scoring functions in industry and academia on both prediction of binding affinity and “native” binding pose. For the target COX-2 of nonsteroidal anti-inflammatory drugs, SPA successfully discriminates the drugs from the diversity set, and the selective drugs from non-selective drugs. The remarkable performance demonstrates that SPA has significant potential applications in identifying lead compounds for drug discovery

Structural Discrimination of Robustness in Transcriptional Feedforward Loops for Pattern Formation

Signaling pathways are interconnected to regulatory circuits for sensing the environment and expressing the appropriate genetic profile. In particular, gradients of diffusing molecules (morphogens) determine cell fate at a given position, dictating development and spatial organization. The feedforward loop (FFL) circuit is among the simplest genetic architectures able to generate one-stripe patterns by operating as an amplitude detection device, where high output levels are achieved at intermediate input ones. Here, using a heuristic optimization-based approach, we dissected the design space containing all possible topologies and parameter values of the FFL circuits. We explored the ability of being sensitive or adaptive to variations in the critical morphogen level where cell fate is switched. We found four different solutions for precision, corresponding to the four incoherent architectures, but remarkably only one mode for adaptiveness, the incoherent type 4 (I4-FFL). We further carried out a theoretical study to unveil the design principle for such structural discrimination, finding that the synergistic action and cooperative binding on the downstream promoter are instrumental to achieve absolute adaptive responses. Subsequently, we analyzed the robustness of these optimal circuits against perturbations in the kinetic parameters and molecular noise, which has allowed us to depict a scenario where adaptiveness, parameter sensitivity and noise tolerance are different, correlated facets of the robustness of the I4-FFL circuit. Strikingly, we showed a strong correlation between the input (environment-related) and the intrinsic (mutation-related) susceptibilities. Finally, we discussed the evolution of incoherent regulations in terms of multifunctionality and robustness

Control of Canalization and Evolvability by Hsp90

Partial reduction of Hsp90 increases expression of morphological novelty in qualitative traits of Drosophila and Arabidopsis, but the extent to which the Hsp90 chaperone also controls smaller and more likely adaptive changes in natural quantitative traits has been unclear. To determine the effect of Hsp90 on quantitative trait variability we deconstructed genetic, stochastic and environmental components of variation in Drosophila wing and bristle traits of genetically matched flies, differing only by Hsp90 loss-of-function or wild-type alleles. Unexpectedly, Hsp90 buffering was remarkably specific to certain normally invariant and highly discrete quantitative traits. Like the qualitative trait phenotypes controlled by Hsp90, highly discrete quantitative traits such as scutellor and thoracic bristle number are threshold traits. When tested across genotypes sampled from a wild population or in laboratory strains, the sensitivity of these traits to many types of variation was coordinately controlled, while continuously variable bristle types and wing size, and critically invariant left-right wing asymmetry, remained relatively unaffected. Although increased environmental variation and developmental noise would impede many types of selection response, in replicate populations in which Hsp90 was specifically impaired, heritability and ‘extrinsic evolvability’, the expected response to selection, were also markedly increased. However, despite the overall buffering effect of Hsp90 on variation in populations, for any particular individual or genotype in which Hsp90 was impaired, the size and direction of its effects were unpredictable. The trait and genetic-background dependence of Hsp90 effects and its remarkable bias toward invariant or canalized traits support the idea that traits evolve independent and trait-specific mechanisms of canalization and evolvability through their evolution of non-linearity and thresholds. Highly non-linear responses would buffer variation in Hsp90-dependent signaling over a wide range, while over a narrow range of signaling near trait thresholds become more variable with increasing probability of triggering all-or-none developmental responses

Robustness in Glyoxylate Bypass Regulation

The glyoxylate bypass allows Escherichia coli to grow on carbon sources with only two carbons by bypassing the loss of carbons as CO2 in the tricarboxylic acid cycle. The flux toward this bypass is regulated by the phosphorylation of the enzyme isocitrate dehydrogenase (IDH) by a bifunctional kinase–phosphatase called IDHKP. In this system, IDH activity has been found to be remarkably robust with respect to wide variations in the total IDH protein concentration. Here, we examine possible mechanisms to explain this robustness. Explanations in which IDHKP works simultaneously as a first-order kinase and as a zero-order phosphatase with a single IDH binding site are found to be inconsistent with robustness. Instead, we suggest a robust mechanism where both substrates bind the bifunctional enzyme to form a ternary complex