Quasi-steady-state kinetics at enzyme and substrate concentrations in excess of the Michaelis-Menten constant

Theory on the rate equations of Michaelis-Menten type enzyme kinetics with competitive inhibition

We derive approximate expressions for pre- and post-steady state regimes of the velocity-substrate-inhibitor spaces of the Michaelis-Menten enzyme kinetic scheme with fully and partial competitive inhibition. Our refinement over the currently available standard quasi steady state approximation (sQSSA) seems to be valid over wide range of enzyme to substrate and enzyme to inhibitor concentration ratios. Further, we show that the enzyme-inhibitor-substrate system can exhibit temporally well-separated two different steady states with respect to both enzyme-substrate and enzyme-inhibitor complexes under certain conditions. We define the ratios fS = vmax/(KMS + e0) and fI = umax/(KMI + e0) as the acceleration factors with respect to the catalytic conversion of substrate and inhibitor into their respective products. Here KMS and KMI are the Michaelis-Menten parameters associated respectively with the binding of substrate and inhibitor with the enzyme, vmax and umax are the respective maximum reaction velocities and e0, s0, and i0 are total enzyme, substrate and inhibitor levels. When (fS/fI) < 1, then enzyme-substrate complex will show multiple steady states and it reaches the full-fledged steady state only after the depletion of enzyme-inhibitor complex. When (fS/fI) > 1, then the enzyme-inhibitor complex will show multiple steady states and it reaches the full-fledged steady state only after the depletion of enzyme-substrate complex. This multi steady-state behavior especially when (fS/fI) ≠ 1 is the root cause of large amount of error in the estimation of various kinetic parameters of fully and partial competitive inhibition schemes using sQSSA. Remarkably, we show that our refined expressions for the reaction velocities over enzyme-substrate-inhibitor space can control this error more significantly than the currently available sQSSA expressions.

Kinetic Study of the Esterase-like Activity of Albumin following Condensation by Macromolecular Crowding

The ability of bovine serum albumin (BSA) to form condensates in crowded environments has been discovered only recently. Effects of this condensed state on the secondary structure of the protein have already been unraveled as some aging aspects, but the pseudo-enzymatic behavior of condensed BSA has never been reported yet. This article investigates the kinetic profile of para-nitrophenol acetate hydrolysis by BSA in its condensed state with poly(ethylene) glycol (PEG) as the crowding agent. Furthermore, the initial BSA concentration was varied between 0.25 and 1 mM which allowed us to modify the size distribution, the volume fraction, and the partition coefficient (varying from 136 to 180). Hence, the amount of BSA originally added was a simple way to modulate the size and density of the condensates. Compared with dilute BSA, the initial velocity (vi) with condensates was dramatically reduced. From the Michaelis-Menten fits, the extracted Michaelis constant Km and the maximum velocity Vmax decreased in control samples without condensates when the BSA concentration increased, which was attributed to BSA self-oligomerization. In samples containing condensates, the observed vi was interpreted as an effect of diluted BSA remaining in the supernatants and from the condensates. In supernatants, the crowding effect of PEG increased the kcat and catalytic efficiency. Last, Vmax was proportional to the volume fraction of the condensates, which could be controlled by varying its initial concentration. Hence, the major significance of this article is the control of the size and volume fraction of albumin condensates, along with their kinetic profile using liquid-liquid phase separation.

Validity conditions of approximations for a target-mediated drug disposition model: A novel first-order approximation and its comparison to other approximations

Target-mediated drug disposition (TMDD) is a phenomenon characterized by a drug's high-affinity binding to a target molecule, which significantly influences its pharmacokinetic profile within an organism. The comprehensive TMDD model delineates this interaction, yet it may become overly complex and computationally demanding in the absence of specific concentration data for the target or its complexes. Consequently, simplified TMDD models employing quasi-steady state approximations (QSSAs) have been introduced; however, the precise conditions under which these models yield accurate results require further elucidation. Here, we establish the validity of three simplified TMDD models: the Michaelis-Menten model reduced with the standard QSSA (mTMDD), the QSS model reduced with the total QSSA (qTMDD), and a first-order approximation of the total QSSA (pTMDD). Specifically, we find that mTMDD is applicable only when initial drug concentrations substantially exceed total target concentrations, while qTMDD can be used for all drug concentrations. Notably, pTMDD offers a simpler and faster alternative to qTMDD, with broader applicability than mTMDD. These findings are confirmed with antibody-drug conjugate real-world data. Our findings provide a framework for selecting appropriate simplified TMDD models while ensuring accuracy, potentially enhancing drug development and facilitating safer, more personalized treatments.

Beyond the Michaelis-Menten: Bayesian Inference for Enzyme Kinetic Analysis

Although the Michaelis-Menten (MM) rate law has been widely used to estimate enzyme kinetic parameters, it works only under the condition of extremely low enzyme concentration. Furthermore, even when this condition is satisfied, parameter estimation is often imprecise due to the parameter identifiability issue. To overcome these limitations of the canonical approach to enzyme kinetics, we developed a Bayesian approach based on a modified form of the MM rate law, which is derived with the total quasi-steady state approximation. Here, we illustrate how to perform the Bayesian inference for the progress curve assay with our user-friendly computational R package. We also describe an optimal experimental design for the progress curve assay, with which enzyme kinetic parameters can be accurately and precisely estimated from minimal measurements of the progress curves.

3D modelling of drug-coated balloons for the treatment of calcified superficial femoral arteries

Background/Objectives

Drug-coated balloon therapy for diseased superficial femoral arteries remains controversial. Despite its clinical relevance, only a few computational studies based on simplistic two-dimensional models have been proposed to investigate this endovascular therapy to date. This work addresses the aforementioned limitation by analyzing the drug transport and kinetics occurring during drug-coated balloon deployment in a three-dimensional geometry.

Methods

An idealized three-dimensional model of a superficial femoral artery presenting with a calcific plaque and treated with a drug-coated balloon was created to perform transient mass transport simulations. To account for the transport of drug (i.e. paclitaxel) released by the device, a diffusion-reaction equation was implemented by describing the drug bound to specific intracellular receptors through a non-linear, reversible reaction. The following features concerning procedural aspects, pathologies and modelling assumptions were investigated: (i) balloon application time (60–180 seconds); (ii) vessel wall composition (healthy vs. calcified wall); (iii) sequential balloon application; and (iv) drug wash-out by the blood stream vs. coating retention, modeled as exponential decay.

Results

The balloon inflation time impacted both the free and specifically-bound drug concentrations in the vessel wall. The vessel wall composition highly affected the drug concentrations. In particular, the specifically-bound drug concentration was four orders of magnitude lower in the calcific compared with healthy vessel wall portions, primarily as a result of reduced drug diffusion. The sequential application of two drug-coated balloons led to modest differences (~15%) in drug concentration immediately after inflation, which became negligible within 10 minutes. The retention of the balloon coating increased the drug concentration in the vessel wall fourfold.

Conclusions

The overall findings suggest that paclitaxel kinetics may be affected not only by the geometrical and compositional features of the vessel treated with the drug-coated balloon, but also by balloon design characteristics and procedural aspects that should be carefully considered.

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.

Explicit Treatment of Non-Michaelis-Menten and Atypical Kinetics in Early Drug Discovery*

Biological systems are highly regulated. They are also highly resistant to sudden perturbations enabling them to maintain the dynamic equilibrium essential to sustain life. This robustness is conferred by regulatory mechanisms that influence the activity of enzymes/proteins within their cellular context to adapt to changing environmental conditions. However, the initial rules governing the study of enzyme kinetics were mostly tested and implemented for cytosolic enzyme systems that were easy to isolate and/or recombinantly express. Moreover, these enzymes lacked complex regulatory modalities. Now, with academic labs and pharmaceutical companies turning their attention to more-complex systems (for instance, multiprotein complexes, oligomeric assemblies, membrane proteins and post-translationally modified proteins), the initial axioms defined by Michaelis-Menten (MM) kinetics are rendered inadequate, and the development of a new kind of kinetic analysis to study these systems is required. This review strives to present an overview of enzyme kinetic mechanisms that are atypical and, oftentimes, do not conform to the classical MM kinetics. Further, it presents initial ideas on the design and analysis of experiments in early drug-discovery for such systems, to enable effective screening and characterisation of small-molecule inhibitors with desirable physiological outcomes.

Triple Strategies to Improve Oral Bioavailability by Fabricating Coamorphous Forms of Ursolic Acid with Piperine: Enhancing Water-Solubility, Permeability, and Inhibiting Cytochrome P450 Isozymes

As a BCS IV drug, ursolic acid (UA) has low oral bioavailability mainly because of its poor aqueous solubility/dissolution, poor permeability, and metabolism by cytochrome P450 (CYP) isozymes, such as CYP3A4. Most UA preparations demonstrated a much higher dissolution than that of its crystalline form yet a low drug concentration in plasma due to their lower consideration or evaluation for the permeability and metabolism issues. In the current study, a supramolecular coamorphous system of UA with piperine (PIP) was prepared and characterized by powder X-ray diffraction, differential scanning calorimetry, and scanning electron microscopy. In comparison to crystalline UA and UA in physical mixture, such coamorphous system enhanced solubility (5.3-7-fold in the physiological solution) and dissolution (7-8-fold in the physiological solution within 2 h) of UA and exhibited excellent physical stability under 90-day storage conditions. More importantly, the pharmacokinetic study of coamorphous UA in rats exhibited 5.8-fold and 2.47-fold improvement in AUC0-∞ value, respectively, compared with its free and mixed crystalline counterparts. In order to further explore the mechanism of such improvement, the molecular interactions of a coamorphous system in the solid state were investigated. Fourier transform infrared spectroscopy, solid-state 13C nuclear magnetic resonance spectroscopy, and density functional theory modeling suggested that intermolecular hydrogen bonds with strong interactions newly formed between UA and PIP after coamorphization. The in vitro permeability studies across Caco-2 cell monolayer and metabolism studies by rat hepatic microsomes indicated that free PIP significantly increased the permeability of UA and inhibited the enzymatic metabolism of UA by CYP3A4. However, PIP in the coamorphous combination exhibited a much lower level in the bioenhancing than its free form arising from the synchronized dissolution characteristic of the preparation (only 60% of PIP released in comparison to its free counterpart in 2 h). The in situ loop study in rats proposed that the acid-sensitive dissolution in the stomach of the coamorphous preparation helped to improve the effective free drug concentration, thereby facilitating PIP to play its role in bioenhancing. The current study offers an exploratory strategy to overcome poor solubility/dissolution, poor permeability, and metabolism by cytochrome P450 isozymes of the BCS IV drug to improve its oral bioavailability.

Misuse of the Michaelis-Menten rate law for protein interaction networks and its remedy

For over a century, the Michaelis-Menten (MM) rate law has been used to describe the rates of enzyme-catalyzed reactions and gene expression. Despite the ubiquity of the MM rate law, it accurately captures the dynamics of underlying biochemical reactions only so long as it is applied under the right condition, namely, that the substrate is in large excess over the enzyme-substrate complex. Unfortunately, in circumstances where its validity condition is not satisfied, especially so in protein interaction networks, the MM rate law has frequently been misused. In this review, we illustrate how inappropriate use of the MM rate law distorts the dynamics of the system, provides mistaken estimates of parameter values, and makes false predictions of dynamical features such as ultrasensitivity, bistability, and oscillations. We describe how these problems can be resolved with a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA). Furthermore, we show that the tQSSA can be used for accurate stochastic simulations at a lower computational cost than using the full set of mass-action rate laws. This review describes how to use quasi-steady state approximations in the right context, to prevent drawing erroneous conclusions from in silico simulations.

The quasi-steady-state approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetics

In this work, we revisit the scaling analysis and commonly accepted conditions for the validity of the standard, reverse and total quasi-steady-state approximations through the lens of dimensional Tikhonov-Fenichel parameters and their respective critical manifolds. By combining Tikhonov-Fenichel parameters with scaling analysis and energy methods, we derive improved upper bounds on the approximation error for the standard, reverse and total quasi-steady-state approximations. Furthermore, previous analyses suggest that the reverse quasi-steady-state approximation is only valid when initial enzyme concentrations greatly exceed initial substrate concentrations. However, our results indicate that this approximation can be valid when initial enzyme and substrate concentrations are of equal magnitude. Using energy methods, we find that the condition for the validity of the reverse quasi-steady-state approximation is far less restrictive than was previously assumed, and we derive a new "small" parameter that determines the validity of this approximation. In doing so, we extend the established domain of validity for the reverse quasi-steady-state approximation. Consequently, this opens up the possibility of utilizing the reverse quasi-steady-state approximation to model enzyme catalyzed reactions and estimate kinetic parameters in enzymatic assays at much lower enzyme to substrate ratios than was previously thought. Moreover, we show for the first time that the critical manifold of the reverse quasi-steady-state approximation contains a singular point where normal hyperbolicity is lost. Associated with this singularity is a transcritical bifurcation, and the corresponding normal form of this bifurcation is recovered through scaling analysis.

An alternative approach to Michaelis-Menten kinetics that is based on the renormalization group

We apply to Michaelis-Menten kinetics an alternative approach to the study of Singularly Perturbed Differential Equations, that is based on the Renormalization Group (SPDERG). To this aim, we first rebuild the perturbation expansion for Michaelis-Menten kinetics, beyond the standard Quasi-Steady-State Approximation (sQSSA), determining the 2nd order contributions to the inner solutions, that are presented here for the first time to our knowledge. Our main result is that the SPDERG 2nd order uniform approximations reproduce the numerical solutions of the original problem in a better way than the known results of the perturbation expansion, even in the critical matching region. Indeed, we obtain analytical results nearly indistinguishable from the numerical solutions of the original problem in a large part of the whole relevant time window, even in the case in which the kinetic constants produce an expansion parameter value as large as ɛ=0.5.

Role of Autoregulation and Relative Synthesis of Operon Partners in Alternative Sigma Factor Networks

Despite the central role of alternative sigma factors in bacterial stress response and virulence their regulation remains incompletely understood. Here we investigate one of the best-studied examples of alternative sigma factors: the σ^B network that controls the general stress response of Bacillus subtilis to uncover widely relevant general design principles that describe the structure-function relationship of alternative sigma factor regulatory networks. We show that the relative stoichiometry of the synthesis rates of σ^B, its anti-sigma factor RsbW and the anti-anti-sigma factor RsbV plays a critical role in shaping the network behavior by forcing the σ^B network to function as an ultrasensitive negative feedback loop. We further demonstrate how this negative feedback regulation insulates alternative sigma factor activity from competition with the housekeeping sigma factor for RNA polymerase and allows multiple stress sigma factors to function simultaneously with little competitive interference.

Understanding the regulation of bacterial stress response holds the key to tackling the problems of emerging resistance to anti-bacteria’s and antibiotics. To this end, here we study one of the longest serving model systems of bacterial stress response: the σ^B pathway of Bacillus subtilis. The sigma factor σ^B controls the general stress response of Bacillus subtilis to a variety of stress conditions including starvation, antibiotics and harmful environmental perturbations. Recent studies have demonstrated that an increase in stress triggers pulsatile activation of σ^B. Using mathematical modeling we identify the core structural design feature of the network that are responsible for its pulsatile response. We further demonstrate how the same core design features are common to a variety of stress response pathways. As a result of these features, cells can respond to multiple simultaneous stresses without interference or competition between the different pathways.

Representação do efeito de inibição enzimática reversível para o modelo cinético de Michaelis-Menten no estado transiente

ResumoOs processos enzimáticos que seguem o modelo cinético de Michaelis-Menten foram estudados a partir de diferentes propostas para descrever a etapa de inibição reversível. As propostas de inibição foram comparadas a partir de um processo genérico, onde as constantes cinéticas receberam valores unitários e o valor numérico da concentração de substrato foi dez (10) vezes superior ao valor numérico da concentração de enzima. Para cada proposta de modelo de inibição foram obtidas soluções numéricas a partir de sistema não linear de equações diferenciais ordinárias, gerando gráficos que apresentaram, separadamente, a variação das concentrações da enzima, dos complexos enzimáticos, do substrato e do produto da reação. Foi obtido um modelo, dentre as propostas avaliadas, com desempenho indicando comportamento similar ao verificado no modelo clássico de Michaelis-Menten, onde o complexo de reação é rapidamente formado e, ao longo do processo, decai até tender a zero. Em contrapartida, diferentemente do modelo clássico, na nova proposta de modelo o efeito de inibição começa em zero e, ao longo do processo, tende ao valor nominal da concentração inicial da enzima. Tais respostas mostraram-se válidas para valores distintos de concentração de enzima e de tempo de processo, mostrando robustez e indicando uma tendência do somatório do substrato e do produto atingir o valor nominal da concentração inicial do substrato ao longo do tempo de processamento.

A generalised enzyme kinetic model for predicting the behaviour of complex biochemical systems

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We propose the dQSSA model as a novel way of modelling complex biological networks.

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No low enzyme concentration assumption, covering more biological settings.

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Reduces the number of parameters, which simplifies optimisation.

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dQSSA was validated both in silico and in vitro.

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Both biochemical and signalling pathways can be modelled accurately and simply.

Quasi steady-state enzyme kinetic models are increasingly used in systems modelling. The Michaelis Menten model is popular due to its reduced parameter dimensionality, but its low-enzyme and irreversibility assumption may not always be valid in the in vivo context. Whilst the total quasi-steady state assumption (tQSSA) model eliminates the reactant stationary assumptions, its mathematical complexity is increased. Here, we propose the differential quasi-steady state approximation (dQSSA) kinetic model, which expresses the differential equations as a linear algebraic equation. It eliminates the reactant stationary assumptions of the Michaelis Menten model without increasing model dimensionality. The dQSSA was found to be easily adaptable for reversible enzyme kinetic systems with complex topologies and to predict behaviour consistent with mass action kinetics in silico. Additionally, the dQSSA was able to predict coenzyme inhibition in the reversible lactate dehydrogenase enzyme, which the Michaelis Menten model failed to do. Whilst the dQSSA does not account for the physical and thermodynamic interactions of all intermediate enzyme-substrate complex states, it is proposed to be suitable for modelling complex enzyme mediated biochemical systems. This is due to its simpler application, reduced parameter dimensionality and improved accuracy.

Zero-order ultrasensitivity: a study of criticality and fluctuations under the total quasi-steady state approximation in the linear noise regime

Zero-order ultrasensitivity (ZOU) is a long known and interesting phenomenon in enzyme networks. Here, a substrate is reversibly modified by two antagonistic enzymes (a 'push-pull' system) and the fraction in modified state undergoes a sharp switching from near-zero to near-unity at a critical value of the ratio of the enzyme concentrations, under saturation conditions. ZOU and its extensions have been studied for several decades now, ever since the seminal paper of Goldbeter and Koshland (1981); however, a complete probabilistic treatment, important for the study of fluctuations in finite populations, is still lacking. In this paper, we study ZOU using a modular approach, akin to the total quasi-steady state approximation (tQSSA). This approach leads to a set of Fokker-Planck (drift-diffusion) equations for the probability distributions of the intermediate enzyme-bound complexes, as well as the modified/unmodified fractions of substrate molecules. We obtain explicit expressions for various average fractions and their fluctuations in the linear noise approximation (LNA). The emergence of a 'critical point' for the switching transition is rigorously established. New analytical results are derived for the average and variance of the fractional substrate concentration in various chemical states in the near-critical regime. For the total fraction in the modified state, the variance is shown to be a maximum near the critical point and decays algebraically away from it, similar to a second-order phase transition. The new analytical results are compared with existing ones as well as detailed numerical simulations using a Gillespie algorithm.

Perturbation theory in the catalytic rate constant of the Henri-Michaelis-Menten enzymatic reaction

The Henry-Michaelis-Menten (HMM) mechanism of enzymatic reaction is studied by means of perturbation theory in the reaction rate constant k (2) of product formation. We present analytical solutions that provide the concentrations of the enzyme (E), the substrate (S), as well as those of the enzyme-substrate complex (C), and the product (P) as functions of time. For k (2) small compared to k (-1), we properly describe the entire enzymatic activity from the beginning of the reaction up to longer times without imposing extra conditions on the initial concentrations E ( o ) and S ( o ), which can be comparable or much different.

Stat5 signaling specifies basal versus stress erythropoietic responses through distinct binary and graded dynamic modalities

Stat5 signaling in erythroblasts can assume either a binary, low-intensity form, essential for basal erythropoiesis, or a graded, high-intensity response, restricted to early erythroblasts and to erythropoietic stress.

Erythropoietin (Epo)-induced Stat5 phosphorylation (p-Stat5) is essential for both basal erythropoiesis and for its acceleration during hypoxic stress. A key challenge lies in understanding how Stat5 signaling elicits distinct functions during basal and stress erythropoiesis. Here we asked whether these distinct functions might be specified by the dynamic behavior of the Stat5 signal. We used flow cytometry to analyze Stat5 phosphorylation dynamics in primary erythropoietic tissue in vivo and in vitro, identifying two signaling modalities. In later (basophilic) erythroblasts, Epo stimulation triggers a low intensity but decisive, binary (digital) p-Stat5 signal. In early erythroblasts the binary signal is superseded by a high-intensity graded (analog) p-Stat5 response. We elucidated the biological functions of binary and graded Stat5 signaling using the EpoR-HM mice, which express a “knocked-in” EpoR mutant lacking cytoplasmic phosphotyrosines. Strikingly, EpoR-HM mice are restricted to the binary signaling mode, which rescues these mice from fatal perinatal anemia by promoting binary survival decisions in erythroblasts. However, the absence of the graded p-Stat5 response in the EpoR-HM mice prevents them from accelerating red cell production in response to stress, including a failure to upregulate the transferrin receptor, which we show is a novel stress target. We found that Stat5 protein levels decline with erythroblast differentiation, governing the transition from high-intensity graded signaling in early erythroblasts to low-intensity binary signaling in later erythroblasts. Thus, using exogenous Stat5, we converted later erythroblasts into high-intensity graded signal transducers capable of eliciting a downstream stress response. Unlike the Stat5 protein, EpoR expression in erythroblasts does not limit the Stat5 signaling response, a non-Michaelian paradigm with therapeutic implications in myeloproliferative disease. Our findings show how the binary and graded modalities combine to generate high-fidelity Stat5 signaling over the entire basal and stress Epo range. They suggest that dynamic behavior may encode information during STAT signal transduction.

Hormone signaling through the erythropoietin (Epo) pathway is required both for the continuous replacement of red blood cells (RBCs) that are lost through aging (a process known as "basal erythropoiesis") and to boost tissue oxygen when bleeding, in anemia or at high altitude ("stress erythropoiesis"). A key challenge lies in understanding how extracellular Epo concentration is translated into different intracellular signals that promote transcription of proteins that are specific to basal versus stress erythropoiesis. Binding of Epo to its receptor EpoR on the surface of an erythroblast (the precursors of RBCs) triggers the addition of phosphates to a target protein Stat5; the phosphorylated Stat5 becomes activated and induces transcription. We show that the dynamic properties of the Stat5 activation signal convey additional information that specifies either basal or stress responses. During basal conditions, the Stat5 signal is low and binary in nature—an on/off switch-like response. Stress, on the other hand, triggers a distinct Stat5 response consisting of a highintensity signal that increases in a graded fashion with rising Epo concentration. We found that a mouse bearing a truncated EpoR is restricted to the low-intensity binary Stat5 signal and correspondingly fails to initiate stress erythropoiesis. Ultimately, it is the Stat5 protein level in erythroblasts that determines their ability to generate the high-intensity graded Stat5 signal in response to high Epo. These findings have therapeutic potential: targeting Stat5's high-intensity graded signal may inhibit its aberrant function in blood cell cancers without affecting its important binary response in normal cells.

Bell-shaped and ultrasensitive dose-response in phosphorylation-dephosphorylation cycles: the role of kinase-phosphatase complex formation

Background

Phosphorylation-dephosphorylation cycles (PDCs) mediated by kinases and phosphatases are common in cellular signalling. Kinetic modelling of PDCs has shown that these systems can exhibit a variety of input-output (dose-response) behaviors including graded response, ultrasensitivity and bistability. In addition to proteins, there are a class of lipids known as phosphoinositides (PIs) that can be phosphorylated. Experimental studies have revealed the formation of an antagonistic kinase-phosphatase complex in regulation of phosphorylation of PIs. However, the functional significance of this type of complex formation is not clear.

Results

We first revisit the basic PDC and show that partial asymptotic phosphorylation of substrate limits ultrasensitivity. Also, substrate levels are changed one can obtain non-monotonic bell-shaped dose-response curves over a narrow range of parameters. Then we extend the PDC to include kinase-phosphatase complex formation. We report the possibility of robust bell-shaped dose-response for a specific class of the model with complex formation. Also, we show that complex formation can produce ultrasensitivity outside the Goldbeter-Koshland zero-order ultrasensitivity regime through a mechanism similar to competitive inhibition between an enzyme and its inhibitor.

Conclusions

We conclude that the novel PDC module studied here exhibits new dose-response behaviour. In particular, we show that the bell-shaped response could result in transient phosphorylation of substrate. We discuss the relevance of this result in the context of experimental observations on PI regulation in endosomal trafficking.

Molecular mechanochemistry: low force switch slows enzymatic cleavage of human type I collagen monomer

In vertebrate animals, fibrillar collagen accumulates, organizes, and persists in structures which resist mechanical force. This antidissipative behavior is possibly due to a mechanochemical force-switch which converts collagen from enzyme-susceptible to enzyme-resistant. Degradation experiments on native tissue and reconstituted fibrils suggest that collagen/enzyme kinetics favor the retention of loaded collagen. We used a massively parallel, single molecule, mechanochemical reaction assay to demonstrate that the effect is derivative of molecular mechanics. Tensile loads higher than 3 pN dramatically reduced (10×) the enzymatic degradation rate of recombinant human type I collagen monomers by Clostridium histolyticum compared to unloaded controls. Because bacterial collagenase accesses collagen at multiple sites and is an aggressive cleaver of the collagen triple helical domain, the results suggest that collagen molecular architecture is generally more stable when mechanically strained in tension. Thus the tensile mechanical state of collagen monomers is likely to be correlated to their longevity in tissues. Further, strain-actuated molecular stability of collagen may constitute the fundamental basis of a smart structural mechanism which enhances the ability of animals to place, retain, and load-optimize material in the path of mechanical forces.

Nanostructure of the fibrin clot

The nanostructure of the fibrin fibers in fibrin clots is investigated by using spectrometry and small angle x-ray scattering measurements. First, an autocoherent analysis of the visible light spectra transmitted through formed clots is demonstrated to provide robust measurements of both the radius and density of the fibrin fibers. This method is validated via comparison with existing small-angle and dynamic light-scattering data. The complementary use of small angle x-ray scattering spectra and light spectrometry unambiguously shows the disjointed nature of the fibrin fibers. Indeed, under quasiphysiological conditions, the fibers are approximately one-half as dense as their crystalline fiber counterparts. Further, although the fibers are locally crystalline, they appear to possess a lateral fractal structure.

Endosomal receptor kinetics determine the stability of intracellular growth factor signalling complexes

There is an emerging paradigm that growth factor signalling continues in the endosome and that cell response to a growth factor is defined by the integration of cell surface and endosomal events. As activated receptors in the endosome are exposed to a different set of binding partners, they probably elicit differential signals compared with when they are at the cell surface. As such, complete appreciation of growth factor signalling requires understanding of growth factor-receptor binding and trafficking kinetics both at the cell surface and in endosomes. Growth factor binding to surface receptors is well characterized, and endosomal binding is assumed to follow surface kinetics if one accounts for changes in pH. Yet, specific binding kinetics within the endosome has not been examined in detail. To parse the factors governing the binding state of endosomal receptors we analysed a whole-cell mathematical model of epidermal growth factor receptor trafficking and binding. We discovered that the stability of growth factor-receptor complexes within endosomes is governed by three primary independent factors: the endosomal dissociation constant, total endosomal volume and the number of endosomal receptors. These factors were combined into a single dimensionless parameter that determines the endosomal binding state of the growth factor-receptor complex and can distinguish different growth factors from each other and different cell states. Our findings indicate that growth factor binding within endosomal compartments cannot be appreciated solely on the basis of the pH-dependence of the dissociation constant and that the concentration of receptors in the endosomal compartment must also be considered.