A study of statistical error in isothermal titration calorimetry

In isothermal titration calorimetry (ITC), the two main sources of random (statistical) error are associated with the extraction of the heat q from the measured temperature changes and with the delivery of metered volumes of titrant. The former leads to uncertainty that is approximately constant and the latter to uncertainty that is proportional to q. The role of these errors in the analysis of ITC data by nonlinear least squares is examined for the case of 1:1 binding, M+X right arrow over left arrow MX. The standard errors in the key parameters-the equilibrium constant Ko and the enthalpy DeltaHo-are assessed from the variance-covariance matrix computed for exactly fitting data. Monte Carlo calculations confirm that these "exact" estimates will normally suffice and show further that neglect of weights in the nonlinear fitting can result in significant loss of efficiency. The effects of the titrant volume error are strongly dependent on assumptions about the nature of this error: If it is random in the integral volume instead of the differential volume, correlated least-squares is required for proper analysis, and the parameter standard errors decrease with increasing number of titration steps rather than increase.     

Calorimetric vs. van't Hoff binding enthalpies from isothermal titration calorimetry: Ba2+-crown ether complexation

The 1:1 complexation reaction between Ba(2+) and 18-crown-6 ether is re-examined using isothermal titration calorimetry (ITC), with the goal of clarifying previously reported discrepancies between reaction enthalpies estimated directly (calorimetric) and indirectly, from the temperature dependence of the reaction equilibrium constant K (van't Hoff). The ITC thermograms are analyzed using three different non-linear fit models based on different assumptions about the data error: constant, proportional to the heat and proportional but correlated. The statistics of the fitting indicate a preference for the proportional error model, in agreement with expectations for the conditions of the experiment, where uncertainties in the delivered titrant volume should dominate. With attention to proper procedures for propagating statistical error in the van't Hoff analysis, the differences between Delta H(cal) and Delta H(vH) are deemed statistically significant. In addition, statistically significant differences are observed for the Delta H(cal) estimates obtained for two different sources of Ba(2+), BaCl(2) and Ba(NO(3))(2). The effects are tentatively attributed to deficiencies in the standard procedure in ITC of subtracting a blank obtained for pure titrant from the thermogram obtained for the sample.     

A novel spectroscopic titration method for determining the dissociation constant and stoichiometry of protein-ligand complex.

We offer a new titration protocol for determining the dissociation constant and binding stoichiometry of protein-ligand complex, detectable by spectroscopic methods. This approach neither is limited to the range of protein or ligand concentrations employed during titration experiment nor relies on precise determinations of the titration "endpoint," i.e., the maximal signal changes upon saturation of protein by ligand (or vice versa). In this procedure, a fixed concentration of protein (or ligand) is titrated by increasing volumes of a stock ligand (or protein) solution, and the changes in the spectroscopic signal are recorded after each addition of the titrant. The signal for interaction between protein and ligand first increases, reaches a maximum value, and then starts decreasing due to dilution effect. The volume of the titrant required to achieve the maximum signal changes is utilized to calculate the dissociation constant and the binding stoichiometry of the protein-ligand complex according to the theoretical relationships developed herein. This procedure has been tested for the interaction of avidin with a chromophoric biotin analogue, 2-(4'-hydroxyazobenzene)benzoic acid by following the absorption signal of their interaction at 500 nm. The widespread applicability of this procedure to protein-ligand complexes detected by other spectroscopic techniques and its advantages over conventional methods are discussed.     

Modeling and analysis of biopathways dynamics

Cellular processes are governed and coordinated by a multitude of biopathways. A pathway can be viewed as a complex network of biochemical reactions. The dynamics of this network largely determines the functioning of the pathway. Hence the modeling and analysis of biochemical networks dynamics is an important problem and is an active area of research. Here we review quantitative models of biochemical networks based on ordinary differential equations (ODEs). We mainly focus on the parameter estimation and sensitivity analysis problems and survey the current methods for tackling them. In this context we also highlight a recently developed probabilistic approximation technique using which these two problems can be considerably simplified.     

A structured approach for the engineering of biochemical network models, illustrated for signalling pathways

Quantitative models of biochemical networks (signal transduction cascades, metabolic pathways, gene regulatory circuits) are a central component of modern systems biology. Building and managing these complex models is a major challenge that can benefit from the application of formal methods adopted from theoretical computing science. Here we provide a general introduction to the field of formal modelling, which emphasizes the intuitive biochemical basis of the modelling process, but is also accessible for an audience with a background in computing science and/or model engineering. We show how signal transduction cascades can be modelled in a modular fashion, using both a qualitative approach--qualitative Petri nets, and quantitative approaches--continuous Petri nets and ordinary differential equations (ODEs). We review the major elementary building blocks of a cellular signalling model, discuss which critical design decisions have to be made during model building, and present a number of novel computational tools that can help to explore alternative modular models in an easy and intuitive manner. These tools, which are based on Petri net theory, offer convenient ways of composing hierarchical ODE models, and permit a qualitative analysis of their behaviour. We illustrate the central concepts using signal transduction as our main example. The ultimate aim is to introduce a general approach that provides the foundations for a structured formal engineering of large-scale models of biochemical networks.     

Application of Petri net based analysis techniques to signal transduction pathways

Background  

Signal transduction pathways are usually modelled using classical quantitative methods, which are based on ordinary differential equations (ODEs). However, some difficulties are inherent in this approach. On the one hand, the kinetic parameters involved are often unknown and have to be estimated. With increasing size and complexity of signal transduction pathways, the estimation of missing kinetic data is not possible. On the other hand, ODEs based models do not support any explicit insights into possible (signal-) flows within the network. Moreover, a huge amount of qualitative data is available due to high-throughput techniques. In order to get information on the systems behaviour, qualitative analysis techniques have been developed. Applications of the known qualitative analysis methods concern mainly metabolic networks. Petri net theory provides a variety of established analysis techniques, which are also applicable to signal transduction models. In this context special properties have to be considered and new dedicated techniques have to be designed.     

The development of a continuous isothermal titration calorimetric method for equilibrium studies

A continuous isothermal titration calorimetry (cITC) method for microcalorimeters has been developed. The method is based on continuous slow injection of a titrant into the calorimetric vessel. The experimental time for a cITC binding experiment is 12-20 min and the number of data points obtained is on the order of 1000. This gives an advantage over classical isothermal titration calorimetry (ITC) binding experiments that need 60-180 min to generate 20-30 data points. The method was validated using two types of calorimeters, which differ in calorimetric principle, geometry, stirring, and way of delivering the titrant into the calorimetric vessel. Two different experimental systems were used to validate the method: the binding of Ba(2+) to 18-crown-6 and the binding of cytidine 2'-monophosphate to RNAse A. Both systems are used as standard test systems for titration calorimetry. Computer simulations show that the dynamic range for determination of equilibrium constants can be increased by three orders of magnitude compared to that of classical ITC, making it possible to determine high affinities. Simulations also show an improved possibility to elucidate the actual binding model from cITC data. The simulated data demonstrate that cITC makes it easier to discriminate between different thermodynamic binding models due to the higher density of data points obtained from one experiment.     

Designing isothermal titration calorimetry experiments for the study of 1:1 binding: problems with the "standard protocol"

Literature recommendations for designing isothermal titration calorimetry (ITC) experiments to study 1:1 binding, M+X -->/<-- MX, are not consistent and have persisted through time with little quantitative justification. In particular, the "standard protocol" employed by most workers involves 20 to 30 injections of titrant to a final titrant/titrand mole ratio (R(m)) of ~ 2-a scheme that can be far from optimal and can needlessly limit applicability of the ITC technique. These deficiencies are discussed here along with other misconceptions. Whether a specific binding process can be studied by ITC is determined less by c (the product of binding constant K and titrand concentration [M](0)) than by the total detectable heat q(tot) and the extent to which M can be converted to MX. As guidelines, with 90% conversion to MX, K can be estimated within 5% over the range 10 to 10(8)M(-1) when q(tot)/σ(q)≈700, where σ(q) is the standard deviation for estimation of q. This ratio drops to ~150 when the stoichiometry parameter n is treated as known. A computer application for modeling 1:1 binding yields realistic estimates of parameter standard errors for use in protocol design and feasibility assessment.     

Mathematical simulation and analysis of cellular metabolism and regulation.

MOTIVATION: A better understanding of the biological phenomena observed in cells requires the creation and analysis of mathematical models of cellular metabolism and physiology. The formulation and study of such models must also be simplified as far as possible to cope with the increasing complexity demanded and exponential accumulation of the metabolic reconstructions computed from sequenced genomes. RESULTS: A mathematical simulation workbench, DBsolve, has been developed to simplify the derivation and analysis of mathematical models. It combines: (i) derivation of large-scale mathematical models from metabolic reconstructions and other data sources; (ii) solving and parameter continuation of non-linear algebraic equations (NAEs), including metabolic control analysis; (iii) solving the non-linear stiff systems of ordinary differential equations (ODEs); (iv) bifurcation analysis of ODEs; (v) parameter fitting to experimental data or functional criteria based on constrained optimization. The workbench has been successfully used for dynamic metabolic modeling of some typical biochemical networks (Dolgacheva et al., Biochemistry (Moscow), 6, 1063-1068, 1996; Goldstein and Goryanin, Mol. Biol. (Moscow), 30, 976-983, 1996), including microbial glycolytic pathways, signal transduction pathways and receptor-ligand interactions. AVAILABILITY: DBsolve 5. 00 is freely available from http://websites.ntl.com/ approximately igor.goryanin. CONTACT: gzz78923@ggr.co.uk     

Symmetries of S-systems.

An S-system is a set of first-order nonlinear differential equations that all have the same structure: The derivative of a variable is equal to the difference of two products of power-law functions. S-systems have been used as models for a variety of problems, primarily in biology. In addition, S-systems possess the interesting property that large classes of differential equations can be recast exactly as S-systems, a feature that has been proven useful in statistics and numerical analysis. Here, simple criteria are introduced that determine whether an S-system possesses certain types of symmetries and how the underlying transformation groups can be constructed. If a transformation group exists, families of solutions can be characterized, the number of S-system equations necessary for solution can be reduced, and some boundary value problems can be reduced to initial value problems.     

Statistical error in isothermal titration calorimetry: variance function estimation from generalized least squares

The method of generalized least squares (GLS) is used to assess the variance function for isothermal titration calorimetry (ITC) data collected for the 1:1 complexation of Ba(2+) with 18-crown-6 ether. In the GLS method, the least squares (LS) residuals from the data fit are themselves fitted to a variance function, with iterative adjustment of the weighting function in the data analysis to produce consistency. The data are treated in a pooled fashion, providing 321 fitted residuals from 35 data sets in the final analysis. Heteroscedasticity (nonconstant variance) is clearly indicated. Data error terms proportional to q(i) and q(i)/v are well defined statistically, where q(i) is the heat from the ith injection of titrant and v is the injected volume. The statistical significance of the variance function parameters is confirmed through Monte Carlo calculations that mimic the actual data set. For the data in question, which fall mostly in the range of q(i)=100-2000 microcal, the contributions to the data variance from the terms in q(i)(2) typically exceed the background constant term for q(i)>300 microcal and v<10 microl. Conversely, this means that in reactions with q(i) much less than this, heteroscedasticity is not a significant problem. Accordingly, in such cases the standard unweighted fitting procedures provide reliable results for the key parameters, K and DeltaH(degrees) and their statistical errors. These results also support an important earlier finding: in most ITC work on 1:1 binding processes, the optimal number of injections is 7-10, which is a factor of 3 smaller than the current norm. For high-q reactions, where weighting is needed for optimal LS analysis, tips are given for using the weighting option in the commercial software commonly employed to process ITC data.     

On the use of qualitative reasoning to simulate and identify metabolic pathways

MOTIVATION: Perhaps the greatest challenge of modern biology is to develop accurate in silico models of cells. To do this we require computational formalisms for both simulation (how according to the model the state of the cell evolves over time) and identification (learning a model cell from observation of states). We propose the use of qualitative reasoning (QR) as a unified formalism for both tasks. The two most commonly used alternative methods of modelling biochemical pathways are ordinary differential equations (ODEs), and logical/graph-based (LG) models. RESULTS: The QR formalism we use is an abstraction of ODEs. It enables the behaviour of many ODEs, with different functional forms and parameters, to be captured in a single QR model. QR has the advantage over LG models of explicitly including dynamics. To simulate biochemical pathways we have developed 'enzyme' and 'metabolite' QR building blocks that fit together to form models. These models are finite, directly executable, easy to interpret and robust. To identify QR models we have developed heuristic chemoinformatics graph analysis and machine learning procedures. The graph analysis procedure is a series of constraints and heuristics that limit the number of ways metabolites can combine to form pathways. The machine learning procedure is generate-and-test inductive logic programming. We illustrate the use of QR for modelling and simulation using the example of glycolysis. AVAILABILITY: All data and programs used are available on request.     

The use of isothermal titration calorimetry to unravel chemotactic signalling mechanisms

Chemotaxis is based on the action of chemosensory pathways and is typically initiated by the recognition of chemoeffectors at chemoreceptor ligand-binding domains (LBD). Chemosensory signalling is highly complex; aspect that is not only reflected in the intricate interaction between many signalling proteins but also in the fact that bacteria frequently possess multiple chemosensory pathways and often a large number of chemoreceptors, which are mostly of unknown function. We review here the usefulness of isothermal titration calorimetry (ITC) to study this complexity. ITC is the gold standard for studying binding processes due to its precision and sensitivity, as well as its capability to determine simultaneously the association equilibrium constant, enthalpy change and stoichiometry of binding. There is now evidence that members of all major LBD families can be produced as individual recombinant proteins that maintain their ligand-binding properties. High-throughput screening of these proteins using thermal shift assays offer interesting initial information on chemoreceptor ligands, providing the basis for microcalorimetric analyses and microbiological experimentation. ITC has permitted the identification and characterization of many chemoreceptors with novel specificities. This ITC-based approach can also be used to identify signal molecules that stimulate members of other families of sensor proteins.     

Estimating parameters for generalized mass action models using constraint propagation

As modern molecular biology moves towards the analysis of biological systems as opposed to their individual components, the need for appropriate mathematical and computational techniques for understanding the dynamics and structure of such systems is becoming more pressing. For example, the modeling of biochemical systems using ordinary differential equations (ODEs) based on high-throughput, time-dense profiles is becoming more common-place, which is necessitating the development of improved techniques to estimate model parameters from such data. Due to the high dimensionality of this estimation problem, straight-forward optimization strategies rarely produce correct parameter values, and hence current methods tend to utilize genetic/evolutionary algorithms to perform non-linear parameter fitting. Here, we describe a completely deterministic approach, which is based on interval analysis. This allows us to examine entire sets of parameters, and thus to exhaust the global search within a finite number of steps. In particular, we show how our method may be applied to a generic class of ODEs used for modeling biochemical systems called Generalized Mass Action Models (GMAs). In addition, we show that for GMAs our method is amenable to the technique in interval arithmetic called constraint propagation, which allows great improvement of its efficiency. To illustrate the applicability of our method we apply it to some networks of biochemical reactions appearing in the literature, showing in particular that, in addition to estimating system parameters in the absence of noise, our method may also be used to recover the topology of these networks.     

Isothermal titration calorimetry of protein-protein interactions.

The interaction of biologicalmacromolecules, whether protein-DNA, antibody-antigen, hormone-receptor, etc., illustrates the complexity and diversity of molecular recognition. The importance of such interactions in the immune response, signal transduction cascades, and gene expression cannot be overstated. It is of great interest to determine the nature of the forces that stabilize the interaction. The thermodynamics of association are characterized by the stoichiometry of the interaction (n), the association constant (K(a)), the free energy (DeltaG(b)), enthalpy (DeltaH(b)), entropy (DeltaS(b)), and heat capacity of binding (DeltaC(p)). In combination with structural information, the energetics of binding can provide a complete dissection of the interaction and aid in identifying the most important regions of the interface and the energetic contributions. Various indirect methods (ELISA, RIA, surface plasmon resonance, etc.) are routinely used to characterize biologically important interactions. Here we describe the use of isothermal titration calorimetry (ITC) in the study of protein-protein interactions. ITC is the most quantitative means available for measuring the thermodynamic properties of a protein-protein interaction. ITC measures the binding equilibrium directly by determining the heat evolved on association of a ligand with its binding partner. In a single experiment, the values of the binding constant (K(a)), the stoichiometry (n), and the enthalpy of binding (DeltaH(b)) are determined. The free energy and entropy of binding are determined from the association constant. The temperature dependence of the DeltaH(b) parameter, measured by performing the titration at varying temperatures, describes the DeltaC(p) term. As a practical application of the method, we describe the use of ITC to study the interaction between cytochrome c and two monoclonal antibodies.     

Determination of molecular parameters by fitting sedimentation data to finite-element solutions of the Lamm equation.

A method for fitting experimental sedimentation velocity data to finite-element solutions of various models based on the Lamm equation is presented. The method provides initial parameter estimates and guides the user in choosing an appropriate model for the analysis by preprocessing the data with the G(s) method by van Holde and Weischet. For a mixture of multiple solutes in a sample, the method returns the concentrations, the sedimentation (s) and diffusion coefficients (D), and thus the molecular weights (MW) for all solutes, provided the partial specific volumes (v) are known. For nonideal samples displaying concentration-dependent solution behavior, concentration dependency parameters for s(sigma) and D(delta) can be determined. The finite-element solution of the Lamm equation used for this study provides a numerical solution to the differential equation, and does not require empirically adjusted correction terms or any assumptions such as infinitely long cells. Consequently, experimental data from samples that neither clear the meniscus nor exhibit clearly defined plateau absorbances, as well as data from approach-to-equilibrium experiments, can be analyzed with this method with enhanced accuracy when compared to other available methods. The nonlinear least-squares fitting process was accomplished by the use of an adapted version of the "Doesn't Use Derivatives" nonlinear least-squares fitting routine. The effectiveness of the approach is illustrated with experimental data obtained from protein and DNA samples. Where applicable, results are compared to methods utilizing analytical solutions of approximated Lamm equations.