Photosynthesis-related quantities for education and modeling

A quantitative understanding of the photosynthetic machinery depends largely on quantities, such as concentrations, sizes, absorption wavelengths, redox potentials, and rate constants. The present contribution is a collection of numbers and quantities related mainly to photosynthesis in higher plants. All numbers are taken directly from a literature or database source and the corresponding reference is provided. The numerical values, presented in this paper, provide ranges of values, obtained in specific experiments for specific organisms. However, the presented numbers can be useful for understanding the principles of structure and function of photosynthetic machinery and for guidance of future research.     

Testing a model for the dynamics of actin structures with biological parameter values

A simple mathematical model for the dynamics of network-bundle transitions in actin filaments has been previously proposed and some of its mathematical properties have been described. Other models in this class have since been considered and investigated mathematically. In this paper, we have made the first steps in connecting parameters in the model with biologically measurable quantities such as published values of rate constants for filament-crosslinker association. We describe how this connection was made, and give some preliminary numerical simulation results for the behavior of the model under biologically realistic parameter regimes. A key result is that filament length influences the bundle-network transition.     

Thermodynamic and Kinetic Analyses of DNA Triplex Formation: Application of Filter-Binding Assay

Abstract

We have developed a simple and efficient method for studying equilibrium thermodynamics and kinetics of DNA triplex formation, which utilizes a filter-binding procedure. The application of this method to the triplex formation between a double-stranded homopurine-homopyrimidine and a single-stranded homopyrimidine oligonucleotides has demonstrated its ability in the quantitative estimation of equilibrium binding constants and rate constants under various conditions. Thus, this simple method can serve as a powerful tool for the systematic analysis of sequence and environmental effects on the equilibrium and kinetic quantities in the triplex formation.     

Kinetics of unfolding and refolding of proteins. I. Mathematical analysis

A mathematical method for the phenomenological analysis of unimolecular reaction kinetics is presented. Starting from the solution to the differential equations of the system, equations have been developed to relate the unknown parameters inherent to any reaction mechanism, such as the microscopic rate constants for individual reaction steps, to the experimental quantities obtained from the rate of change of average physical properties of the system. Provided that kinetic measurements can be made in both forward and reverse directions, all unknown parameters can be evaluated for many simple mechanisms, and for some of them characteristic relations between observable quantities can be established, which can serve as criteria for immediate rejection. The results will be applied in the following papers to experimental studies of denaturation and renaturation of several proteins.     

Anion-mediated iron release from transferrins. The kinetic and mechanistic model for N-lobe of ovotransferrin

Iron release process of ovotransferrin N-lobe (N-oTf) to anion/chelators has been resolved using kinetic and mechanistic approach. The iron release kinetics of N-oTf were measured at the endosomal pH of 5.6 with three different anions such as nitrilotriacetate, pyrophosphate, and sulfate using stopped flow spectrofluorimetric method, all yielding clear biphasic progress curves. The two observed rate constants and the corresponding amplitudes obtained from the double exponential curve fit to the biphasic curves varied depending on the type and concentration of anions. Several possible models for the iron release kinetic mechanism were examined on the basis of a newly introduced quantitative equation. Results from the curve fitting analyses were consistent with a dual pathway mechanism that includes the competitive iron release from two different protein states, namely, X and Y, with the respective first order rate constants of K(1) and K(2) (X, domain closed holo N-oTf; Y, anion induced different conformer of holo N-oTf). The reversible interconversions of X to Y and Y to X are driven by the second order rate constant k(3) and the first order rate constant K(4), respectively. The obtained rate constants were greatly variable for the three anions depending on the synergistic or nonsynergistic nature. In the light of the anion-binding sites of N-oTf located crystallographically, the compatible mechanistic model that includes competitive anion binding to the iron coordination sites and to a specific anion site is suggested for the dual pathway iron release mechanism.     

Ultrasensitive investigations of biological systems by fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) extracts information about molecular dynamics from the tiny fluctuations that can be observed in the emission of small ensembles of fluorescent molecules in thermodynamic equilibrium. Employing a confocal setup in conjunction with highly dilute samples, the average number of fluorescent particles simultaneously within the measurement volume (approximately 1 fl) is minimized. Among the multitude of chemical and physical parameters accessible by FCS are local concentrations, mobility coefficients, rate constants for association and dissociation processes, and even enzyme kinetics. As any reaction causing an alteration of the primary measurement parameters such as fluorescence brightness or mobility can be monitored, the application of this noninvasive method to unravel processes in living cells is straightforward. Due to the high spatial resolution of less than 0.5 microm, selective measurements in cellular compartments, e.g., to probe receptor-ligand interactions on cell membranes, are feasible. Moreover, the observation of local molecular dynamics provides access to environmental parameters such as local oxygen concentrations, pH, or viscosity. Thus, this versatile technique is of particular attractiveness for researchers striving for quantitative assessment of interactions and dynamics of small molecular quantities in biologically relevant systems.     

Multiscale brain modelling

A central difficulty of brain modelling is to span the range of spatio-temporal scales from synapses to the whole brain. This paper overviews results from a recent model of the generation of brain electrical activity that incorporates both basic microscopic neurophysiology and large-scale brain anatomy to predict brain electrical activity at scales from a few tenths of a millimetre to the whole brain. This model incorporates synaptic and dendritic dynamics, nonlinearity of the firing response, axonal propagation and corticocortical and corticothalamic pathways. Its relatively few parameters measure quantities such as synaptic strengths, corticothalamic delays, synaptic and dendritic time constants, and axonal ranges, and are all constrained by independent physiological measurements. It reproduces quantitative forms of electroencephalograms seen in various states of arousal, evoked response potentials, coherence functions, seizure dynamics and other phenomena. Fitting model predictions to experimental data enables underlying physiological parameters to be inferred, giving a new non-invasive window into brain function that complements slower, but finer-resolution, techniques such as fMRI. Because the parameters measure physiological quantities relating to multiple scales, and probe deep structures such as the thalamus, this will permit the testing of a range of hypotheses about vigilance, cognition, drug action and brain function. In addition, referencing to a standardized database of subjects adds strength and specificity to characterizations obtained.     

Determination of kinetic parameters from chemical relaxation data discrimination between coupled two-step binding reactions differing in stoichiometry

The chemical relaxation times of two different two-step equilibrium reactions, characterized by a 1:1 binding process followed by a subsequent rearrangement step and a stepwise 1:2 binding reaction, are analyzed for the purpose of qualitative model discrimination and quantitative determination of kinetic parameters. The equations describing the dependences of the two reciprocal relaxation times on suitable concentrations are given for both models in the general case as well as for four different limiting situations which are characterized by well separated relaxation times. The conditions corresponding to the limiting cases are expressed in terms of strong, weak and no coupling between the two partial equilibrium steps involved in both models. The coupling strength depends on the rate constants as well as on the total concentrations of the reactants. Criteria to discriminate between these two reaction models under defined limiting conditions are developed. In the general case, the product of both reciprocal relaxation times can be used to distinguish both models. If only one relaxation time can be resolved experimentally, it is possible under conditions described to determine only a reduced set of individual rate constants for most of the limiting cases considered. If both relaxation times are observed, all rate constants are determinable in the general case as well as in most of the limiting cases discussed.     

Singular perturbation analysis of cAMP signalling in Dictyostelium discoideum aggregates

In this paper, we use singular perturbation methods to study the structure of travelling waves for some reaction-diffusion models obtained from the Martiel-Goldbeter and Goldbeter-Segel's models of cAMP signalling in Dictyostelium discoideum. As a consequence, we derive analytic formulae for quantities like wave speed, maximum concentration and other magnitudes in terms of the different biochemical constants that appear in the model.     

Fitting yeast and mammalian prion aggregation kinetic data with the Finke-Watzky two-step model of nucleation and autocatalytic growth

Recently, we reported 14 amyloid protein aggregation kinetic data sets that were fit using the "Ockham's razor"/minimalistic Finke-Watzky (F-W) two-step model of slow nucleation (A --> B, rate constant k 1) and fast autocatalytic growth (A + B --> 2B, rate constant k 2), yielding quantitative (average) rate constants for nucleation ( k 1) and growth ( k 2), where A is the monomeric protein and B is the polymeric protein [Morris, A. M., et al. (2008) Biochemistry 47, 2413-2427]. Herein, we apply the F-W model to 27 representative prion aggregation kinetic data sets obtained from the literature. Each prion data set was successfully fit with the F-W model, including three different yeast prion proteins (Sup35p, Ure2p, and Rnq1p) as well as mouse and human prions. These fits yield the first quantitative rate constants for the steps of nucleation and growth in prion aggregation. Examination of a Sup35p system shows that the same rate constants are obtained for nucleation and for growth within experimental error, regardless of which of six physical methods was used, a unique set of important control experiments in the protein aggregation literature. Also provided herein are analyses of several factors influencing the aggregation of prions such as glutamine/asparagine rich regions and the number of oligopeptide repeats in the prion domain. Where possible, verification or refutation of previous correlations to glutamine/asparagine regions, or the number of repeat sequences, in literature aggregation kinetics is given in light of the quantitative rate constants obtained herein for nucleation and growth during prion aggregation. The F-W model is then contrasted to four literature mechanisms that address the molecular picture of prion transmission and propagation. Key limitations of the F-W model are listed to prevent overinterpretation of the data being analyzed, limitations that derive ultimately from the model's simplicity. Finally, possible avenues of future research are suggested.     

Scintillation proximity assay as a high-throughput method to identify slowly dissociating nonpeptide ligand binding to the GnRH receptor

Many nonpeptide antagonists of the gonadotropin-releasing hormone (GnRH) receptor, as well as other drug targets, possess a broad range of dissociation kinetic rate constants. Current methods to accurately define kinetic rate parameters such as K(on) and K(off) are time and labor intensive, prompting the development of a screening assay to identify slowly dissociating compounds for follow-up rate constant determination. The authors measured inhibition binding constants (K(i)) for GnRH receptor antagonists after 30 min and 10 h of incubation and observed several compounds with markedly decreased K(i) values over time (Ki(30 min)/Ki(10 h) > 6). They used scintillation proximity assay technology to perform these binding experiments because this homogeneous assay does not have a fixed termination end point as does filtration binding, permitting successive readings to be taken from the same assay plate over an extended period of time. They also used a quantitative method of kinetic rate analysis to confirm that a large disparity between a compound's K(i) value at 30 min and 10 h could identify compounds that dissociate slowly. Thus, the K(i) ratio can be used to screen for and select compounds to test using more quantitative, albeit lower throughput methods to accurately define kinetic rate constants.     

Reflections on post-OECD eutrophication models

In recent years a contrast has become apparent between oversimplified models (regression lines) relating primary production in lakes to phosphate loadings, and complicated conceptual models, depending on the rate constants of the most important processes. In the latter, rate constants from the literature are used. Most modellers assume that such constants are valid everywhere and always, but very little research concerning their validity has been carried out.It is generally believed that Smith's equation for photosynthesis and Monod's for bacterial growth under nutrient limited conditions are proven conceptual models — and that Monod's equation may be used for algal growth as well.     

Computation of diagnostic data from coronary sinus pressure: a comparison between two possible models

We have evaluated two mathematical models to describe the increase in coronary sinus pressure (CSP) following pressure controlled intermittent coronary sinus occlusion (PICSO). The models are evaluated and compared on the basis of human and canine data. Both models were fitted by non-linear least squares algorithms. Next, derived quantities, such as plateau, rise-time and mean integral of the coronary sinus pressure were calculated from the model parameters. Corresponding quantities for the two models were compared with regard to mean values, rate of successful calculation and specific features characterizing the human or canine case. One model was found to be superior for investigational purposes. The other model was found to be more stable in critical situations and is therefore suggested for usage in closed loop regulation of PICSO. Physiologically, the differences in mean values of the derived quantities between the two models were found to be negligible. The formal statistical significance of the differences is but a consequence of the large number of PICSO cycles analysed.     

Four-state models and regulation of contraction of smooth muscle. I. Physical considerations, stability, and solutions.

Cyclic four-state models are frequently used in biology to represent a variety of molecular behaviors. A common experimental strategy to test such models is to follow the behavior of the real system after some of the rate constants are changed in a stepwise manner. We analyze the mathematical behavior of a simple example of such a model applicable to the regulation of contraction of smooth muscle, but our results apply in general to any linear, cyclic four-state model. We discuss detailed balance and requirements for linearity. We find that the only way to have sustained oscillations is for the rate constants of the model to be themselves oscillatory. We state conditions for decaying oscillations and find that in models that do not follow strictly first-order kinetics and do not satisfy detailed balance, these conditions can hold. We show analytically that the response of any state to step changes in the rate constants is the sum of three weighted exponentials plus a constant term, the steady-state value. We provide explicit expressions for the time dependence of all state variables. We discuss a simple way to use these results to obtain numerical solutions in cases where the rate constants change in an arbitrary way.     

Physical modulation of intracellular signaling processes by locational regulation.

Recent observations in the field of signal transduction suggest that where a protein is located within a cell can be as important as its activity measured in solution for activation of its downstream pathway. The physical organization of the cell can provide an additional layer of control upon the chemical reaction networks that govern ultimately perceived signals. Using the cytosol and plasma membrane as relevant compartmental distinctions, we analyze the effect of relocation on the rate of association with a membrane-associated target. We quantify this effect as an enhancement factor E in terms of measurable parameters such as the number of available targets, molecular diffusivities, and intrinsic reaction rate constants. We then employ two simple yet relevant example models to illustrate how relocation can affect the dynamics of signal transduction pathways. The temporal profiles and phase behavior of these models are investigated. We also relate experimentally observable aspects of signal transduction such as peak activation and the relative time scales of stimulus and response to quantitative aspects of the relocation mechanisms in our models. In our example schemes, nearly complete relocation of the cytosolic species in the signaling pair is required to generate meaningful activation of the model pathways when the association rate enhancement factor E is as low as 10; when E is 100 or greater, only a small fraction of the protein must be relocated.     

Modeling vital rates improves estimation of population projection matrices

Population projection matrices are commonly used by ecologists and managers to analyze the dynamics of stage-structured populations. Building projection matrices from data requires estimating transition rates among stages, a task that often entails estimating many parameters with few data. Consequently, large sampling variability in the estimated transition rates increases the uncertainty in the estimated matrix and quantities derived from it, such as the population multiplication rate and sensitivities of matrix elements. Here, we propose a strategy to avoid overparameterized matrix models. This strategy involves fitting models to the vital rates that determine matrix elements, evaluating both these models and ones that estimate matrix elements individually with model selection via information criteria, and averaging competing models with multimodel averaging. We illustrate this idea with data from a population of Silene acaulis (Caryophyllaceae), and conduct a simulation to investigate the statistical properties of the matrices estimated in this way. The simulation shows that compared with estimating matrix elements individually, building population projection matrices by fitting and averaging models of vital-rate estimates can reduce the statistical error in the population projection matrix and quantities derived from it.     

Robustness Analysis of Stochastic Biochemical Systems

We propose a new framework for rigorous robustness analysis of stochastic biochemical systems that is based on probabilistic model checking techniques. We adapt the general definition of robustness introduced by Kitano to the class of stochastic systems modelled as continuous time Markov Chains in order to extensively analyse and compare robustness of biological models with uncertain parameters. The framework utilises novel computational methods that enable to effectively evaluate the robustness of models with respect to quantitative temporal properties and parameters such as reaction rate constants and initial conditions. We have applied the framework to gene regulation as an example of a central biological mechanism where intrinsic and extrinsic stochasticity plays crucial role due to low numbers of DNA and RNA molecules. Using our methods we have obtained a comprehensive and precise analysis of stochastic dynamics under parameter uncertainty. Furthermore, we apply our framework to compare several variants of two-component signalling networks from the perspective of robustness with respect to intrinsic noise caused by low populations of signalling components. We have successfully extended previous studies performed on deterministic models (ODE) and showed that stochasticity may significantly affect obtained predictions. Our case studies demonstrate that the framework can provide deeper insight into the role of key parameters in maintaining the system functionality and thus it significantly contributes to formal methods in computational systems biology.     

Maximum likelihood estimation of molecular motor kinetics from staircase dwell-time sequences

Molecular motors, such as kinesin, myosin, or dynein, convert chemical energy into mechanical energy by hydrolyzing ATP. The mechanical energy is used for moving in discrete steps along the cytoskeleton and carrying a molecular load. High resolution single molecule recordings of motor steps appear as a stochastic sequence of dwells, resembling a staircase. Staircase data can also be obtained from other molecular machines such as F1 -ATPase, RNA polymerase, or topoisomerase. We developed a maximum likelihood algorithm that estimates the rate constants between different conformational states of the protein, including motor steps. We model the motor with a periodic Markov model that reflects the repetitive chemistry of the motor step. We estimated the kinetics from the idealized dwell-sequence by numerical maximization of the likelihood function for discrete-time Markov models. This approach eliminates the need for missed event correction. The algorithm can fit kinetic models of arbitrary complexity, such as uniform or alternating step chemistry, reversible or irreversible kinetics, ATP concentration and mechanical force-dependent rates, etc. The method allows global fitting across stationary and nonstationary experimental conditions, and user-defined a priori constraints on rate constants. The algorithm was tested with simulated data, and implemented in the free QuB software.