The Universal Statistical Distributions of the Affinity,Equilibrium Constants,Kinetics and Specificity in Biomolecular Recognition

We uncovered the universal statistical laws for the biomolecular recognition/binding process. We quantified the statistical energy landscapes for binding, from which we can characterize the distributions of the binding free energy (affinity), the equilibrium constants, the kinetics and the specificity by exploring the different ligands binding with a particular receptor. The results of the analytical studies are confirmed by the microscopic flexible docking simulations. The distribution of binding affinity is Gaussian around the mean and becomes exponential near the tail. The equilibrium constants of the binding follow a log-normal distribution around the mean and a power law distribution in the tail. The intrinsic specificity for biomolecular recognition measures the degree of discrimination of native versus non-native binding and the optimization of which becomes the maximization of the ratio of the free energy gap between the native state and the average of non-native states versus the roughness measured by the variance of the free energy landscape around its mean. The intrinsic specificity obeys a Gaussian distribution near the mean and an exponential distribution near the tail. Furthermore, the kinetics of binding follows a log-normal distribution near the mean and a power law distribution at the tail. Our study provides new insights into the statistical nature of thermodynamics, kinetics and function from different ligands binding with a specific receptor or equivalently specific ligand binding with different receptors. The elucidation of distributions of the kinetics and free energy has guiding roles in studying biomolecular recognition and function through small-molecule evolution and chemical genetics.     

Effect of Zn2+ on the thermal denaturation of carboxypeptidase B

A differential scanning calorimetry study on the thermal denaturation of porcine pancreas carboxypeptidase B (in 20 mM pyrophosphate buffer, pH 9.0) has been carried out. The calorimetric transitions have been found to be calorimetrically irreversible and to depend on the Zn2+ concentration in the buffer. The effect of the Zn2+ concentration on the temperatures corresponding to maximum heat capacity appears to conform the dictates of the van't Hoff equation. In spite of this, analysis of the scanning rate effect on the transitions, together with studies on the thermal inactivation kinetics, show that the heat absorption is entirely determined by the rate of formation of the final (irreversibly denatured) state of the protein; therefore, analysis of the calorimetric transitions according to equilibrium thermodynamics models is not permissible. The effect of Zn2+ on the calorimetric transitions can be explained on the basis of a simple kinetic model that does not assume chemical equilibrium to be established between the significantly populated states of the protein.     

Cardiac chemical constant. Part I: Derivation of the characteristic phenomenological equation

A physiochemical parameter is derived and defined as the cardiac chemical equilibrium dissociation constant (K_D), K_D is based upon a phenomenological model in which the cardiac muscle chemical reaction kinetics describe the interconversion between long and short unils (i.e. the individual sarcomere is fully extended or fully contracted). K_D is defined as the ratio of the number of units in the long state to the number of units in the short state. The mathematical development proceeds through four stages: derivation of the governing differential equation during cardiac systole; simplification of the differential equation to describe the cardiac model; determination of the upper and lower limits and average value of Nt (the total number of units in a hypothetical mid-wall circumferential fibre); definition and calculation of the cardiac chemical constant (K_D). K_D is shown to describe a series of equilibrium points throughout cardiac systole. This requires that each mechanical equilibrium state (a series of static, steady-state intervals over time) is also associated with its own specific chemical equilibrium state.     

Thermodynamics and kinetics of formation of the alkaline state of a Lys 79-->Ala/Lys 73-->His variant of iso-1-cytochrome c

The alkaline transition kinetics of a Lys 73-->His (H73) variant of iso-1-cytochrome c are triggered by three ionizable groups [Martinez, R. E., and Bowler, B. E. (2004) J. Am. Chem. Soc. 126, 6751-6758]. To eliminate ambiguities caused by overlapping phases due to formation of the Lys 79 alkaline conformer and proline isomerization associated with the His 73 alkaline conformer, we mutated Lys 79 to Ala in the H73 variant (A79H73). The stability and guanidineHCl m-values of the A79H73 and H73 variants at pH 7.5 are the same. The Ala 79 mutation causes formation of the alkaline conformer to depend on [NaCl]. The salt dependence saturates at 500 mM NaCl, and the thermodynamics of alkaline state formation for the A79H73 and H73 variants become identical. The salt dependence is consistent with loss of an electrostatic contact between Lys 79 and heme propionate D in the A79H73 variant. The kinetics of alkaline state formation for the A79H73 variant support the three trigger group model developed for the H73 variant, with the primary trigger, pK(HL), being ionization of His 73. The low pH ionization, pK(H1), is perturbed by the Ala 79 mutation indicating that this ionization is modulated by the buried hydrogen bond network involving heme propionate D. The A79H73 variant has a high spin heme above pH 9 suggesting that the high pH ionization, pK(H2), involves a high spin heme conformer. The proline isomerization phase is modulated by both pK(HL) and pK(H2) indicating that it is sensitive to protein conformation.     

Mathematical modelling of glycolysis and adenine nucleotide metabolism of human erythrocytes. I. Reaction-kinetic statements, analysis of in vivo state and determination of starting conditions for in vitro experiments

A mathematical model of energy metabolism of human red cells is presented, which includes besides the glycolytic reactions the adenine nucleotide metabolism. The model is based on the network of chemical reactions, the thermodynamic equilibrium constants of fast reversible reactions and on the kinetic equations for irreversible enzyme reactions. The model consists of a system of 16 differential equations and allows the mathematical evaluation of metabolic levels in the steady state of energy metabolism corresponding to the in vivo state erythrocytes with the kinetic data for the enzymes derived from in vitro experiments. The dependence of the levels of metabolites in the steady state on the activity of some enzymes is analysed to characterize the regulatory properties of the system. The comparison of the steady state levels of the model with experimental data makes it possible to estimate values of some controversial enzyme parameters. Estimates of the kinetic parameters of the following intracellular processes are presented: 1) rate constant of AMP-phosphatase, 2) maximum rate of adenylate deaminase, 3) activity of adenine phosphoribosylpyrophosphate transferase and 4) adenosine transport through the cell membrane. The simulation of the preparatory phase before incubation of erythrocytes indicates, that the model also permits to compute the time course of changes of levels of metabolites. To solve the initial problem the stiff differential equation system is integrated numerically by an efficient program without the application of the quasi-steady-state approximation.     

Pathways of the rise and decay of the M photointermediate(s) of bacteriorhodopsin

The photocycle of bacteriorhodopsin (BR) was studied at alkaline pH with a gated multichannel analyzer, in order to understand the origins of kinetic complexities in the rise and decay of the M intermediate. The results indicate that the biphasic rise and decay kinetics are unrelated to a photoreaction of the N intermediate of the BR photocycle, proposed earlier by others [Kouyama et al. (1988) Biochemistry 27, 5855-5863]. Rather, under conditions where N did not accumulate in appreciable amounts (high pH, low salt concentration), they were accounted for by conventional kinetic schemes. These contained reversible interconversions, either M in equilibrium with N in one of two parallel photocycles or L in equilibrium with as well as M in equilibrium with N in a single photocycle. Monomeric BR also showed these kinetic complications. Conditions were then created where N accumulated in a photo steady state (high pH, high salt concentration, background illumination). The apparent increase in the proportion of the slow M decay component by the background illumination could be quantitatively accounted for with the single photocycle model, by the mixing of the relaxation of the background light induced photo steady state with the inherent kinetics of the photocycle. Postulating a new M intermediate which is produced by the photoreaction of N was neither necessary nor warranted by the data. The difference spectra suggested instead that absorption of light by N generates only one intermediate, observable between 100 ns and 1 ms, which absorbs near 610 nm. Thus, the photoreaction of N resembles in some respects that of BR containing 13-cis-retinal.     

Modelling circadian rhythms of protein KaiA, KaiB and KaiC interactions in cyanobacteria

Cyanobacteria are the simplest organisms known that exhibit circadian rhythms. The mechanism of circadian rhythm generation in cyanobacteria is different from eukaryotes. Based on the recent experiments about the interaction of KaiA, KaiB, and KaiC proteins with the generation of circadian rhythms in vitro, we developed a mathematical model to describe post-translational oscillations and the possible chemical reactions involved in the circadian clock mechanism of cyanobacteria. In this model, a series of differential equations, with linear kinetics for binding of proteins, Michaelis - Menten kinetics for enzymatic processes and a term including an explicit delay for dissociation of the KaiA/KaiB/phospho-KaiC complex, are proposed describing the dynamics of the chemistry. It is demonstrated that the mathematical system can lead to circadian oscillation within a range of parameter values.     

Stability of the human respiratory control system II. Analysis of a three-dimensional delay state-space model

A number of mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system. Among them, periodic breathing (including Cheyne-Stokes respiration and apneustic breathing) is a collection of regular but involuntary breathing patterns that have important medical implications. The hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of Grodins et al. in the early 1950's [1]. The purpose of this paper is to study the stability characteristics of a feedback control system of five differential equations with delays in both the state and control variables presented by Khoo et al. [4] in 1991 for modeling human respiration. The paper is divided in two parts. Part I studies a simplified mathematical model of two nonlinear state equations modeling arterial partial pressures of O2 and CO2 and a peripheral controller. Analysis was done on this model to illuminate the effect of delay on the stability. It shows that delay dependent stability is affected by the controller gain, compartmental volumes and the manner in which changes in the ventilation rate is produced (i.e., by deeper breathing or faster breathing). In addition, numerical simulations were performed to validate analytical results. Part II extends the model in Part I to include both peripheral and central controllers. This, however, necessitates the introduction of a third state equation modeling CO2 levels in the brain. In addition to analytical studies on delay dependent stability, it shows that the decreased cardiac output (and hence increased delay) resulting from the congestive heart condition can induce instability at certain control gain levels. These analytical results were also confirmed by numerical simulations.     

Temperature sensitivity of organic matter decomposition in the Arrhenius equation: some theoretical considerations

Previous theoretical analyses based on Arrhenius kinetics and thermodynamics have shown that the temperature sensitivity of low-quality substrate is higher than that of high-quality substrate. Because soils store large amounts of low-quality carbon, understanding its response to increasing temperatures will help to predict the response of atmospheric CO₂ to climate change. However, empirical studies do not provide conclusive evidence to corroborate this theoretical argument. Although there are various possible reasons for this disagreement, the theory behind this argument has not been scrutinized carefully. Based on a simple mathematical analysis of the Arrhenius equation it is shown here that low-quality substrates are less temperature sensitive when analyzed in absolute rather than in relative terms, a result that may seem counterintuitive to previous theory. However, this is a paradox intrinsic to the Arrhenius equation and it is often ignored within the ‘quality-temperature’ debate. In fact, different measures commonly used to analyze the temperature sensitivity of different substrates can provide apparently different and contradictory results even though they are based on the same basic principles. Distinguishing between absolute and relative measures of sensitivity is essential for understanding the sensitivity of respiration to environmental change. An analysis of the available empirical evidence on this topic shows that most studies actually agree with the Arrhenius and thermodynamics theory, with less disagreement than previously thought. To address some of the issues identified here, a formal theoretical framework is proposed to study the sensitivity of respiration rates with respect to changes in multiple drivers of decomposition.     

Denatured state is critical in determining the properties of model proteins designed on different folds

The thermodynamics of proteins designed on three common folds (SH3, chymotrypsin inhibitor 2 [CI2], and protein G) is studied with a simplified C(alpha) model and compared with the thermodynamics of proteins designed on random-generated folds. The model allows to design sequences to fold within a dRMSD ranging from 1.2 to 4.2 A from the crystallographic native conformation and to study properties that are hard to be measured experimentally. It is found that the denatured state of all of them is not random but is, to different extents, partially structured. The degree of structure is more abundant for SH3 and protein G, giving rise to a weaker stability but a more efficient folding kinetics than CI2 and, even more, than the random-generated folds. Consequently, the features of the unfolded state seem to be as important in the determination of the thermodynamic properties of these proteins as the features of the native state.     

Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, is presented. Korzuhin's Theorem. which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem ‘for beginners’ is also given.     

A model of tumor growth based on cell cycle kinetics

This work describes a mathematical model of growth based on the kinetics of the cell cycle. A traditional model of the cell cycle has been used, with the addition of a resting (G₀) state from which cells could reenter the reproductive cycle. The model assumes that a growth regulatory substance regulates the transition of cells to and from the resting state. Other transitions between the phases of the cycle were modeled as a first order process. Cell loss is an important feature of growth kinetics, and has been represented by a general but tractable mathematical form. The resulting model forms a system of ordinary nonlinear differential equations. Analytic methods are employed first in the study of this system. Simplifying assumptions regarding cell loss give rise to special cases for which equilibrium solutions can be found. One special case, which assumes first order loss from all cell cycle phases at equal rates, is presented here. For small time values, approximations corresponding to exponential growth were developed. The equations describing an intrinsic growth rate were derived. Simulation methods were used to further characterize the behavior of this model. Parameter values were chosen based on animal tumor cell cycle kinetic data, resulting in a set of 45 model simulations. Several tumor treatment protocols were simulated which illustrated the importance of the intrinsic growth rate and cell loss concepts. Although the qualitative behavior regarding absolute and relative growth is reasonable, this model awaits data for model fitting, parameter estimation, or revision of the equations.     

Approximations and their consequences for dynamic modelling of signal transduction pathways

Signal transduction is the process by which the cell converts one kind of signal or stimulus into another. This involves a sequence of biochemical reactions, carried out by proteins. The dynamic response of complex cell signalling networks can be modelled and simulated in the framework of chemical kinetics. The mathematical formulation of chemical kinetics results in a system of coupled differential equations. Simplifications can arise through assumptions and approximations. The paper provides a critical discussion of frequently employed approximations in dynamic modelling of signal transduction pathways. We discuss the requirements for conservation laws, steady state approximations, and the neglect of components. We show how these approximations simplify the mathematical treatment of biochemical networks but we also demonstrate differences between the complete system and its approximations with respect to the transient and steady state behavior.     

Low level viral persistence after infection with LCMV: a quantitative insight through numerical bifurcation analysis

Many important viruses persist at very low levels in the body in the face of host immunity, and may influence the maintenance of this state of 'infection immunity'. To analyse low level viral persistence in quantitative terms, we use a mathematical model of antiviral cytotoxic T lymphocyte (CTL) response to lymphocytic choriomeningitis virus (LCMV).This model, described by a non-linear system of delay differential equations (DDEs), is studied using numerical bifurcation analysis techniques for DDEs. Domains where low level LCMV coexistence with CTL memory is possible, either as an equilibrium state or an oscillatory pattern, are identified in spaces of the model parameters characterising the interaction between virus and CTL populations. Our analysis suggests that the coexistence of replication competent virus below the conventional detection limit (of about 100 pfu per spleen) in the immune host as an equilibrium state requires the per day relative growth rate of the virus population to decrease at least 5-fold compared to the acute phase of infection. Oscillatory patterns in the dynamics of persisting LCMV and CTL memory, with virus population varying between 1 and 100 pfu per spleen, are possible within quite narrow intervals of the rates of virus growth and precursor CTL population death. Whereas the virus replication rate appears to determine the stability of the low level virus persistence, it does not affect the steady-state level of the viral population, except for very low values.     

The deactivation of Photosystem II in Chlorella as a second-order process

The kinetics of deactivation of the S₃ state in Chlorella have been observed under a variety of conditions. The S₃ state appears to decline in a dark period coming after a sequence of 30 saturating flashes in a second-order reaction, the rate constant of which is 0.132/[S*₃] s⁻¹ and which involves an electron donor, D₁, of concentration 1.25[S*₃] where [S*₃] is the concentration of the S₃ state when the oxygen yield of the light flashes is constant. If a 1 min period of 650 nm illumination is employed after the sequence of flashes, the subsequent S₃ state deactivation kinetics are more complex. There is an initial phase of S₃ state deactivation, accounting for about 35% of the original S₃ state, which is complete in less than 100 ms. The remaining 65% of the S₃ state appears to deactivate in a second-order reaction, the rate constant of which is 1.36/[S*₃] s⁻¹ and which involves an electron donor of initial concentration 0.58[S*₃]. If a 1 min period of 710 nm illumination comes after the 30 flashes, at least 98% of the S₃ state deactivates according to first-order kinetics. It is shown that this can be explained using a second-order model if there is an electron donor present of which the concentration is large compared with [S*₃]. However, S₃ state deactivation observed after 5 min of dark and two saturating flashes can be described neither by a first-order model nor a second-order model. Deactivation of the S₂ state after a 5 min dark period and one saturating flash follows second-order kinetics with a rate constant of 0.2/[S*₃] s⁻¹ and appears to involve an electron donor of initial concentration 1.3[S*₃]. Arguments are presented which tend to rule out the primary electron acceptor to Photosystem II as being any of the electron donors but it appears quite possible that the large plastoquinone pool is involved.     

Kinetics, thermodynamics and evolution of non-native interactions in a protein folding nucleus

A lattice model with side chains was used to investigate protein folding with computer simulations. In this model, we rigorously demonstrate the existence of a specific folding nucleus. This nucleus contains specific interactions not present in the native state that, when weakened, slow folding but do not change protein stability. Such a decoupling of folding kinetics from thermodynamics has been observed experimentally for real proteins. From our results, we conclude that specific non-native interactions in the transition state would give rise to straight phi-values that are negative or larger than unity. Furthermore, we demonstrate that residue Ile 34 in src SH3, which has been shown to be kinetically, but not thermodynamically, important, is universally conserved in proteins with the SH3 fold. This is a clear example of evolution optimizing the folding rate of a protein independent of its stability and function.     

Observation of multiple radical pair states in photosystem 2 reaction centers.

Charge recombination of the primary radical pair in D1/D2 reaction centers from photosystem 2 has been studied by time-resolved fluorescence and absorption spectroscopy. The kinetics of the primary radical pair are multiexponential and exhibit at least two lifetimes of 20 and 52 ns. In addition, a third lifetime of approximately 500 ps also appears to be present. These multiexponential charge-recombination kinetics reflect either different conformational states of D1/D2 reaction centers, with the different conformers exhibiting different radical pair lifetimes, or relaxations in the free energy of the radical pair state. Whichever model is invoked, the free energies of formation of the different radical pair states exhibit a linear temperature dependence from 100 to 220 K, indicating that they are dominated by entropy with negligible enthalpy contributions. These results are in agreement with previous determinations of the thermodynamics that govern primary charge separation in both D1/D2 reaction centers [Booth, P.J., Crystall, B., Giorgi, L. B., Barber, J., Klug, D.R., & Porter, G. (1990) Biochim. Biophys. Acta 1016, 141-152] and reaction centers of purple bacteria [Woodbury, N.W.T., & Parson, W.W. (1984) Biochim. Biophys. Acta 767, 345-361]. It is possible that these observations reflect structural changes that accompanying primary charge separation and assist in stabilization of the radical pair state thus optimizing the efficiency of primary electron transfer.     

Temperature and guanidine hydrochloride dependence of the structural stability of ribonuclease T1.

The thermal unfolding of ribonuclease T1 has been studied by high-sensitivity differential scanning calorimetry as a function of temperature, [GuHCl], and scanning rate. The destabilizing effect of GuHCl has revealed that the kinetics of the unfolding transition become extremely slow as the transition temperature decreases. At pH 5.3 and zero GuHCl, the unfolding transition is centered at 59.1 degrees C; upon increasing the GuHCl concentration, the transition occurs at lower temperatures and exhibits progressively slower kinetics; so, for example, at 3 M GuHCl, the transition temperature is 40.6 degrees C and is characterized by a time constant close to 10 min. Under all conditions studied (pH 5.3, pH 7.0, [GuHCl] < 3 M), the transition is thermodynamically reversible. The slow kinetics of the transition induce significant distortions in the shape of the transition profiles that can be mistakenly interpreted as deviations from a two-state mechanism. Determination of the thermodynamic parameters from the calorimetric data has required the development of an analytical formalism that explicitly includes the thermodynamics as well as the kinetics of the transition. Using this formalism, it is shown that a two-state slow-kinetics model is capable of accurately describing the structural stability of ribonuclease T1 as a function of temperature, GuHCl concentration, and scanning rate. Multidimensional analysis of the calorimetric data has been used to estimate the intrinsic thermodynamic parameters for protein stability, the interaction parameters with GuHCl, and the time constant for the unfolding transition and its temperature dependence.     

Mechanisms of the multiphasic kinetics in the folding and unfolding of globular proteins

The multiphasic kinetics of the protein folding and unfolding processes are examined for a “cluster model” with only two thermodynamically stable macroscopic states, native (N) and denatured (D), which are essentially distributions of microscopic states. The simplest kinetic schemes consistent with the model are: N-(fast) → I-(slow) → D for unfolding and N ← (fast)-D₂ ← (slow)-D₁ for refolding. The fast phase during the unfolding process can be visualized as the redistribution of the native population N to I within its free energy valley. Then, this population crosses over the free energy barrier to the denatured state D in the slow phase. Therefore, the macrostate I is a kinetic intermediate which is not stable at equilibrium. For the refolding process, the initial equilibrium distribution of the denatured state D appears to be separated into D₁ and D₂ in the final condition because of the change in position of the free energy barrier. The fast refolding species D₂ is due to the “leak” from the broadly distributed D state, while the rest is the slow refolding species D₁, which must overpass the free energy barrier to reach N. At an early stage of the folding process the amino acid chain is considered to be composed of several locally ordered regions, which we call clusters, connected by random coil chain parts. Thus, the denatured state contains different sizes and distributions of clusters depending on the external condition. A later stage of the folding process is the association of smaller clusters. The native state is expressed by a maximum-size cluster with possible fluctuation sites reflecting this association. A general discussion is given of the correlation between the kinetics and thermodynamics of proteins from the overall shape of the free energy function. The cluster model provides a conceptual link between the folding kinetics and the structural patterns of globular proteins derived from the X-ray crystallographic data.     

Electron transfer between azurin from Alcaligenes faecalis and cytochrome c551 from Pseudomonas aeruginosa

The electron transfer equilibrium and kinetics between azurin from Alcaligenes faecalis and cytochrome c551 from Pseudomonas aeruginosa have been studied. The equilibrium constant K = ([Cyt(III)] . [Az(I)])/([Cyt(II)] . [Az(II))]) = 0.5 at 25 degrees C is about seven times smaller than that observed between the cytochrome c551 and the titrations confirmed a 43-mV difference between the mid-point potentials of +266 mV and +309 mV for the Alcaligenes and Pseudomonas azurins respectively. The kinetics of the reaction between Alcaligenes azurin and Pseudomonas cytochrome c551 were investigated by the temperature-jump chemical relaxation method. Only a single relaxation mode was observed throughout the range of concentrations and temperatures examined. Thus, the slow relaxation time observed in the reaction between P. aeruginosa azurin and cytochrome c551 is not observed with the Alcaligenes azurin. The simplest mechanism that can therefore be ascribed to the investigated system is: [formula: see text]. This scheme is similar to that proposed earlier for the reaction between P. aeruginosa azurin and cytochrome c551 but does not involve the conformational transition proposed for azurin. The specific rates for the electron transfer are still fast: 1.8 x 10(6) M-1 . s-1 and 3.0 x 10(6) M-1 . s-1 respectively at 25 degrees C.