Systematic search for the rate constants that control the exocytotic process from chromaffin cells by a genetic algorithm

We have recently created a kinetic model that reproduces the dynamics of exocytosis with high accuracy. The reconstruction necessitated a search, in a multi-dimensional parameter space, for 37 parameters that described the system, with no assurance that the parameters, which reconstructed the observations, are a unique set. In the present study, a Genetic Algorithm (GA) was used for a thorough search in the unknown parameter space, using a strategy of gradual increase of the complexity of the analyzed input data. Upon systematic incorporation of one to four measurable parameters, used as input signals for the analysis, the constraint set on the GA search imposed the convergence of the free parameters into a single narrow range. The mean values for each adjustable parameter represent a minimum for the fitness function in the multi-dimensional parameter space. The GA search demonstrates that the parameters that control the kinetics of exocytosis are the rate constants of the steps downstream to synaptotagmin binding, and that the equilibrium constant of the binding of calcium to Munc13 controls the calcium-dependent priming process. Thus, the systematic use of the GA creates a link between specific reactions in the process of exocytosis and experimental phenotypes.     

The feasibility of parameterizing four-state equilibria using relaxation dispersion measurements

Coupled equilibria play important roles in controlling information flow in biochemical systems, including allosteric molecules and multidomain proteins. In the simplest case, two equilibria are coupled to produce four interconverting states. In this study, we assessed the feasibility of determining the degree of coupling between two equilibria in a four-state system via relaxation dispersion measurements. A major bottleneck in this effort is the lack of efficient approaches to data analysis. To this end, we designed a strategy to efficiently evaluate the smoothness of the target function surface (TFS). Using this approach, we found that the TFS is very rough when fitting benchmark CPMG data to all adjustable variables of the four-state equilibria. After constraining a portion of the adjustable variables, which can often be achieved through independent biochemical manipulation of the system, the smoothness of TFS improves dramatically, although it is still insufficient to pinpoint the solution. The four-state equilibria can be finally solved with further incorporation of independent chemical shift information that is readily available. We also used Monte Carlo simulations to evaluate how well each adjustable parameter can be determined in a large kinetic and thermodynamic parameter space and how much improvement can be achieved in defining the parameters through additional measurements. The results show that in favorable conditions the combination of relaxation dispersion and biochemical manipulation allow the four-state equilibrium to be parameterized, and thus coupling strength between two processes to be determined.     

A method for parameter optimization in computational biology

Models in computational biology, such as those used in binding, docking, and folding, are often empirical and have adjustable parameters. Because few of these models are yet fully predictive, the problem may be nonoptimal choices of parameters. We describe an algorithm called ENPOP (energy function parameter optimization) that improves-and sometimes optimizes-the parameters for any given model and for any given search strategy that identifies the stable state of that model. ENPOP iteratively adjusts the parameters simultaneously to move the model global minimum energy conformation for each of m different molecules as close as possible to the true native conformations, based on some appropriate measure of structural error. A proof of principle is given for two very different test problems. The first involves three different two-dimensional model protein molecules having 12 to 37 monomers and four parameters in common. The parameters converge to the values used to design the model native structures. The second problem involves nine bumpy landscapes, each having between 4 and 12 degrees of freedom. For the three adjustable parameters, the globally optimal values are known in advance. ENPOP converges quickly to the correct parameter set.     

Efficient Characterization of Parametric Uncertainty of Complex (Bio)chemical Networks

Parametric uncertainty is a particularly challenging and relevant aspect of systems analysis in domains such as systems biology where, both for inference and for assessing prediction uncertainties, it is essential to characterize the system behavior globally in the parameter space. However, current methods based on local approximations or on Monte-Carlo sampling cope only insufficiently with high-dimensional parameter spaces associated with complex network models. Here, we propose an alternative deterministic methodology that relies on sparse polynomial approximations. We propose a deterministic computational interpolation scheme which identifies most significant expansion coefficients adaptively. We present its performance in kinetic model equations from computational systems biology with several hundred parameters and state variables, leading to numerical approximations of the parametric solution on the entire parameter space. The scheme is based on adaptive Smolyak interpolation of the parametric solution at judiciously and adaptively chosen points in parameter space. As Monte-Carlo sampling, it is “non-intrusive” and well-suited for massively parallel implementation, but affords higher convergence rates. This opens up new avenues for large-scale dynamic network analysis by enabling scaling for many applications, including parameter estimation, uncertainty quantification, and systems design.     

Evolutionary optimization with data collocation for reverse engineering of biological networks

MOTIVATION: Modern experimental biology is moving away from analyses of single elements to whole-organism measurements. Such measured time-course data contain a wealth of information about the structure and dynamic of the pathway or network. The dynamic modeling of the whole systems is formulated as a reverse problem that requires a well-suited mathematical model and a very efficient computational method to identify the model structure and parameters. Numerical integration for differential equations and finding global parameter values are still two major challenges in this field of the parameter estimation of nonlinear dynamic biological systems. RESULTS: We compare three techniques of parameter estimation for nonlinear dynamic biological systems. In the proposed scheme, the modified collocation method is applied to convert the differential equations to the system of algebraic equations. The observed time-course data are then substituted into the algebraic system equations to decouple system interactions in order to obtain the approximate model profiles. Hybrid differential evolution (HDE) with population size of five is able to find a global solution. The method is not only suited for parameter estimation but also can be applied for structure identification. The solution obtained by HDE is then used as the starting point for a local search method to yield the refined estimates.     

Parameterized specification, configuration and execution of data-intensive scientific workflows

Data analysis processes in scientific applications can be expressed as coarse-grain workflows of complex data processing operations with data flow dependencies between them. Performance optimization of these workflows can be viewed as a search for a set of optimal values in a multidimensional parameter space consisting of input performance parameters to the applications that are known to affect their execution times. While some performance parameters such as grouping of workflow components and their mapping to machines do not affect the accuracy of the analysis, others may dictate trading the output quality of individual components (and of the whole workflow) for performance. This paper describes an integrated framework which is capable of supporting performance optimizations along multiple such parameters. Using two real-world applications in the spatial, multidimensional data analysis domain, we present an experimental evaluation of the proposed framework.     

A need for a 'whole-istic functional genomics' approach in complex human diseases: arthritis

'Genomic tools', such as gene/protein chips, single nucleotide polymorphism and haplotype analyses, are empowering us to generate staggering amounts of correlative data, from human/animal genetics and from normal and disease-affected tissues obtained from complex diseases such as arthritis. These tools are transforming molecular biology into a 'data rich' science, with subjects with an '-omic' suffix. These disciplines have to converge and integrate at a systemic level to examine the structure and dynamics of cellular and organismal function ('functionomics') simultaneously, using a multidimensional approach for cells, tissues, organs, rodents and Zebra fish models, which intertwines various approaches and readouts to study the development and homeostasis of a system. In summary, the postgenomic era of functionomics will facilitate narrowing the bridge between correlative data and causative data, thus integrating 'intercoms' of interacting and interdependent disciplines and forming a unified whole.     

Protein structure prediction force fields: Parametrization with quasi-newtonian dynamics

We present an unusual method for parametrizing low-resolution force fields of the type used for protein structure prediction. Force field parameters were-determined by assigning each a fictitious mass and using a quasi-molecular dynamics algorithm in parameter space. The quasi-energy term favored folded native structures and specifically penalized folded nonnative structures. The force field was generated after optimizing less than 70 adjustable parameters, but shows a strong ability to discriminate between native structures and compact misfolded-alternatives. The functional form of the force field was chosen as in molecular mechanics and is not table-driven. It is continuous with continuous derivatives and is thus suitable for use with algorithms such as energy minimization or newtonian dynamics. Proteins 27:367–384, 1997. © 1997 Wiley-Liss, Inc.     

Dimension reduction of multivariate outputs of socio-environmental agent-based models

Often, socio-environmental agent-based models (ABMs) are driven by a host of parameters, and their outputs are similarly multidimensional or vastly high-dimensional. While this complex data and its inter-relationships may be rendered tractable, the task is far from trivial. In this paper, we study the multidimensional outcome space of the socio-environmental, land-use RHEA ABM (Risks and Hedonics in an Empirical Agent-based Land Market), specifically the inter-distances among the outcome measures, and their reducibility using several well-known dimension reduction techniques, variants of multidimensional scalings. In testing the efficacy of several reduction algorithms, we temporally characterize the model’s reducibility while exposing changes in behavior across a wide parameter space that can signal sudden or gradual shifts and possible critical transitions. Our findings reveal that the ABM’s signature reducibility trends exhibit idiosyncrasies and unexpected non-linearity as well as discontinuities. These non-linearities and discontinuities are indicative of both gradual and sudden shifts, signaling potential propensity to internal perturbations induced by parameter settings. Additionally, we related the outcome space via their inter-distances to the multidimensional input parameter space, effectively assessing outcome “reducibility to model controls”. This analysis reveals that the model’s sensitivity to parameters is not only temporally dependent, but also can be partitioned by them, some of which suppress variability in this reduction to model controls, raising questions regarding the extent and structure of endogeneity that yields the distinct temporal trends in the relationship between inputs and outputs and their connections to outcome reducibility.     

Determination of best-fit potential parameters for a reactive force field using a genetic algorithm

The ReaxFF interatomic potential, used for organic materials, involves more than 600 adjustable parameters, the best-fit values of which must be determined for different materials. A new method of determining the set of best-fit parameters for specific molecules containing carbon, hydrogen, nitrogen and oxygen is presented, based on a parameter reduction technique followed by genetic algorithm (GA) minimization. This work has two novel features. The first is the use of a parameter reduction technique to determine which subset of parameters plays a significant role for the species of interest; this is necessary to reduce the optimization space to manageable levels. The second is the application of the GA technique to a complex potential (ReaxFF) with a very large number of adjustable parameters, which implies a large parameter space for optimization. In this work, GA has been used to optimize the parameter set to determine best-fit parameters that can reproduce molecular properties to within a given accuracy. As a test problem, the use of the algorithm has been demonstrated for nitromethane and its decomposition products.     

Redox-linked proton translocation in cytochrome oxidase: the importance of gating electron flow. The effects of slip in a model transducer.

In at least one component of the mitochondrial respiratory chain, cytochrome c oxidase, exothermic electron transfer reactions are used to drive vectorial proton transport against an electrochemical hydrogen ion gradient across the mitochondrial inner membrane. The role of the gating of electrons (the regulation of the rates of electron transfer into and out of the proton transport site) in this coupling between electron transfer and proton pumping has been explored. The approach involves the solution of the steady-state rate equations pertinent to proton pump models which include, to various degrees, the uncoupled (i.e., not linked to proton pumping) electron transfer processes which are likely to occur in any real electron transfer-driven proton pump. This analysis furnishes a quantitative framework for examining the effects of variations in proton binding site pKas and metal center reduction potentials, the relationship between energy conservation efficiency and turnover rate, the conditions for maximum power output or minimum heat production, and required efficiency of the gating of electrons. Some novel conclusions emerge from the analysis, including: An efficient electron transfer-driven proton pump need not exhibit a pH-dependent reduction potential; Very efficient gating of electrons is required for efficient electron transfer driven proton pumping, especially when a reasonable correlation of electron transfer rate and electron transfer exoergonicity is assumed; and A consideration of the importance and possible mechanisms of the gating of electrons suggests that efficient proton pumping by CuA in cytochrome oxidase could, in principle, take place with structural changes confined to the immediate vicinity of the copper ion, while proton pumping by Fea would probably require conformational coupling between the iron and more remote structures in the enzyme. The conclusions are discussed with reference to proton pumping by cytochrome c oxidase, and some possible implications for oxidative phosphorylation are noted.     

Marcus rate theory applied to enzymatic proton transfer

The hydration of CO(2) and the dehydration of HCO(3)(-) catalyzed by the carbonic anhydrases is accompanied by the transfer of protons between solution and the zinc-bound water molecule in the active site. This transfer is facilitated by amino acid residues of the enzyme which act as intramolecular proton shuttles; variants of carbonic anhydrase lacking such shuttle residues are enhanced or rescued in catalysis by intermolecular proton transfer from donors such as imidazole in solution. The resulting rate constants for proton transfer when compared with the values of the pK(a) of the donor and acceptor give Bronsted plots of high curvature. These data are described by Marcus theory which shows an intrinsic barrier for proton transfer from 1 to 2 kcal/mol and work terms or thermodynamic contributions to the free energy of reaction from 4 to10 kcal/mol. The interpretation of these Marcus parameters is discussed in terms of the well-studied pathway of the catalysis and structure of the enzymes.     

Dynamics of the proton transfer reaction on the cytoplasmic surface of bacteriorhodopsin

The cytoplasmic surface of bacteriorhodopsin is characterized by a group of carboxylates that function as a proton attractive domain [Checover, S., Nachliel, E., Dencher, N. A., and Gutman, M. (1997) Biochemistry 36, 13919-13928]. To identify these carboxylates, we selectively mutated them into cysteine residues and monitored the effects of the dynamics of proton transfer between the bulk and the surface of the protein. The measurements were carried out without attachment of a pH-sensor to the cysteine residue, thus avoiding any structural perturbation and change in the surface charge caused by the attachment of a reporter group, and the protein was in its BR state. The purple membranes were suspended in an unbuffered solution of pyranine (8-hydroxypyrene-1,3,6-trisulfonate) and exposed to a train of 1000 laser pulses (2.1 mJ/pulse, lambda = 355 nm, at 10 Hz). The excitation of the dye ejected the hydroxyl's proton, and a few nanoseconds later, a pair of free protons and ground-state pyranine anion was formed. The experimental observation was the dynamics of the relaxation of the system to the prepulse state. The observed signals were reconstructed by a numeric method that replicates the chemical reactions proceeding in the perturbed space. The detailed reconstruction of the measured signal assigned the various proton-binding sites with rate constants for proton binding and proton exchange and the pK values. Comparison of the results obtained by the various mutants indicates that the dominant proton-binding cluster of the wild-type protein consists of D104, E161, and E234. The replacement of D104 or E161 with cysteine lowered the proton binding capacity of the cluster to approximately 60% of that of the native protein. The replacement of E234 with cysteine disrupted the structure of the cluster, causing the two remaining carboxylates to function as isolated residues that do not interact with each other. The possibility of proton transfer between monomers is discussed.     

Minimal models of multidimensional computations

The multidimensional computations performed by many biological systems are often characterized with limited information about the correlations between inputs and outputs. Given this limitation, our approach is to construct the maximum noise entropy response function of the system, leading to a closed-form and minimally biased model consistent with a given set of constraints on the input/output moments; the result is equivalent to conditional random field models from machine learning. For systems with binary outputs, such as neurons encoding sensory stimuli, the maximum noise entropy models are logistic functions whose arguments depend on the constraints. A constraint on the average output turns the binary maximum noise entropy models into minimum mutual information models, allowing for the calculation of the information content of the constraints and an information theoretic characterization of the system's computations. We use this approach to analyze the nonlinear input/output functions in macaque retina and thalamus; although these systems have been previously shown to be responsive to two input dimensions, the functional form of the response function in this reduced space had not been unambiguously identified. A second order model based on the logistic function is found to be both necessary and sufficient to accurately describe the neural responses to naturalistic stimuli, accounting for an average of 93% of the mutual information with a small number of parameters. Thus, despite the fact that the stimulus is highly non-Gaussian, the vast majority of the information in the neural responses is related to first and second order correlations. Our results suggest a principled and unbiased way to model multidimensional computations and determine the statistics of the inputs that are being encoded in the outputs.     

Energy transport mechanism in the form of proton soliton in a one-dimensional hydrogen-bonded polypeptide chain

The dynamics of protons in a one-dimensional hydrogen-bonded (HB) polypeptide chain (PC) is investigated theoretically. A new Hamiltonian is formulated with the inclusion of higher-order molecular interactions between peptide groups (PGs). The wave function of the excitation state of a single particle is replaced by a new wave function of a two-quanta quasi-coherent state. The dynamics is governed by a higher-order nonlinear Schrödinger equation and the energy transport is performed by the proton soliton. A nonlinear multiple-scale perturbation analysis has been performed and the evolution of soliton parameters such as velocity and amplitude is explored numerically. The proton soliton is thermally stable and very robust against these perturbations. The energy transport by the proton soliton is more appropriate to understand the mechanism of energy transfer in biological processes such as muscle contraction, DNA replication, and neuro-electric pulse transfer on biomembranes.     

Hands-on parameter search for neural simulations by a MIDI-controller

Computational neuroscientists frequently encounter the challenge of parameter fitting--exploring a usually high dimensional variable space to find a parameter set that reproduces an experimental data set. One common approach is using automated search algorithms such as gradient descent or genetic algorithms. However, these approaches suffer several shortcomings related to their lack of understanding the underlying question, such as defining a suitable error function or getting stuck in local minima. Another widespread approach is manual parameter fitting using a keyboard or a mouse, evaluating different parameter sets following the users intuition. However, this process is often cumbersome and time-intensive. Here, we present a new method for manual parameter fitting. A MIDI controller provides input to the simulation software, where model parameters are then tuned according to the knob and slider positions on the device. The model is immediately updated on every parameter change, continuously plotting the latest results. Given reasonably short simulation times of less than one second, we find this method to be highly efficient in quickly determining good parameter sets. Our approach bears a close resemblance to tuning the sound of an analog synthesizer, giving the user a very good intuition of the problem at hand, such as immediate feedback if and how results are affected by specific parameter changes. In addition to be used in research, our approach should be an ideal teaching tool, allowing students to interactively explore complex models such as Hodgkin-Huxley or dynamical systems.     

Visualization of multidimensional spectra in flow cytometry.

Flow data from a cell sorter have been processed by hardwired circuits which include amplification, discrimination, coincidence requirements, peak sensing and holding, A-D conversion, and a computerized pulse height analysis with storage of the spectra obtained. Two dimensional spectra can be stored directly in memory, on tape and disk. Three and four parametric cellular events can be recorded on line during the flow measurement in a sequential mode on tape for subsequent recall. Simple processing of these data can be performed for displaying of two dimensional projections from these multidimensional spaces based on threshold conditions for the remaining parameters. Interfaced transmission of the stored data to a large scale computer enables more sophisticated data analysis. Data reduction by means of a multidimensional probability analysis has been carried out in order to transfer the spectra to a computerized picture system for display. This system creates perspective two-dimensional images from a three-dimensional data space. Frequency can be converted into grey levels. Hard copy in color (color as the third dimension and color intensity as frequency) simplifies the visualization of multiparametric flow data sets.     

Understanding the cytochrome c oxidase proton pump: thermodynamics of redox linkage.

The cytochrome c oxidase complex (CcO) catalyzes the four-electron reduction of dioxygen to water by using electrons from ferrocytochrome c. Redox free energy released in this highly exergonic process is utilized to drive the translocation of protons across a transmembrane electrochemical gradient. Although numerous chemical models of proton pumping have been developed, few attempts have been made to explain the stepwise transfer of energy in the context of proposed protein conformational changes. A model is described that seeks to clarify the thermodynamics of the proton pumping function of CcO and that illustrates the importance of electron and proton gating to prevent the occurrence of the more exergonic electron leak and proton slip reactions. The redox energy of the CcO-membrane system is formulated in terms of a multidimensional energy surface projected into two dimensions, a nuclear coordinate associated with electron transfer and a nuclear coordinate associated with elements of the proton pump. This model provides an understanding of how a transmembrane electrochemical gradient affects the efficiency of the proton pumping process. Specifically, electron leak and proton slip reactions become kinetically viable as a result of the greater energy barriers that develop for the desired reactions in the presence of a transmembrane potential.