Fixed-point methods for computing the equilibrium composition of complex biochemical mixtures

An Invitation to Pharmacostatics

Pharmacology, the study of interactions between biological processes and therapeutic agents, is traditionally presented as consisting of two subdisciplines: pharmacokinetics, which is about the distribution and metabolism of drugs in organisms, and pharmacodynamics, which is about the organisms' response to drugs. In discovery-stage pharmacology however, one primary concern is what we call pharmacostatics, the characterization of equilibrium parameters and states of core interactions of physiologic and therapeutic interest. This usually takes the form of studying dose-response curves, without consideration for the relevant qualitative properties of the underlying reaction networks, e.g., the existence, multiplicity and asymptotic stability of steady states. Furthermore, steady-state calculations customarily employ manually derived closed-form expressions based on approximating assumptions. While these formulas may seem adequate most of the time, the assumptions need not apply, and there are genuine though seemingly uncommon cases where this approach is not feasible and/or fails to explain non-monotone dose-response curves. It is this paper's aim to stimulate interest in mathematical problems arising in pharmacostatics. We specifically pose two problems about a particular relevant class of networks of reversible binding reactions. The first problem is to exploit a certain fixed-point formulation of the equilibrium equation to devise an algorithmic method that would be compellingly preferable to current practice in the pharmacostatics context. The second problem is to explicitly anticipate the possibility of non-monotone dose-response curves from network topology. Addressing these problems would positively impact biopharmaceutical research, and they have inherent mathematical interest.

Equilibrium model selection: dTTP induced R1 dimerization

Background

Biochemical equilibria are usually modeled iteratively: given one or a few fitted models, if there is a lack of fit or over fitting, a new model with additional or fewer parameters is then fitted, and the process is repeated. The problem with this approach is that different analysts can propose and select different models and thus extract different binding parameter estimates from the same data. An alternative is to first generate a comprehensive standardized list of plausible models, and to then fit them exhaustively, or semi-exhaustively.

Results

A framework is presented in which equilibriums are modeled as pairs (g, h) where g = 0 maps total reactant concentrations (system inputs) into free reactant concentrations (system states) which h then maps into expected values of measurements (system outputs). By letting dissociation constants K_d be either freely estimated, infinity, zero, or equal to other K_d, and by letting undamaged protein fractions be either freely estimated or 1, many g models are formed. A standard space of g models for ligand-induced protein dimerization equilibria is given. Coupled to an h model, the resulting (g, h) were fitted to dTTP induced R1 dimerization data (R1 is the large subunit of ribonucleotide reductase). Models with the fewest parameters were fitted first. Thereafter, upon fitting a batch, the next batch of models (with one more parameter) was fitted only if the current batch yielded a model that was better (based on the Akaike Information Criterion) than the best model in the previous batch (with one less parameter). Within batches models were fitted in parallel. This semi-exhaustive approach yielded the same best models as an exhaustive model space fit, but in approximately one-fifth the time.

Conclusion

Comprehensive model space based biochemical equilibrium model selection methods are realizable. Their significance to systems biology as mappings of data into mathematical models warrants their development.

A generalized numerical approach to rapid-equilibrium enzyme kinetics: application to 17beta-HSD

A generalized numerical treatment of rapid-equilibrium enzyme kinetics is presented. This new approach relies on automatic computer derivation of the underlying mathematical model (a system of simultaneous nonlinear algebraic equations) from a symbolic representation of the reaction mechanism (a system of biochemical equations) provided by the researcher. The method allows experimental biochemists to analyze initial-rate enzyme kinetic data without having to use any mathematical equations. An illustrative example is based on the inhibition kinetics of 17beta-hydroxysteroid dehydrogenase type 5 by a class of natural compounds. A computer implementation of the new method, a newly modified software package DYNAFIT [Kuzmic, P., 1996. Program DYNAFIT for the analysis of enzyme kinetic data: application to HIV proteinase. Anal. Biochem. 237, 260-273], is freely available to all academic researchers.

Homooligomerization of the cytoplasmic domain of the T cell receptor zeta chain and of other proteins containing the immunoreceptor tyrosine-based activation motif

Antigen receptors on T cells, B cells, mast cells, and basophils all have cytoplasmic domains containing one or more copies of an immunoreceptor tyrosine-based activation motif (ITAM), tyrosine residues of which are phosphorylated upon receptor engagement in an early and obligatory event in the signaling cascade. How clustering of receptor extracellular domains leads to phosphorylation of cytoplasmic domain ITAMs is not known, and little structural or biochemical information is available for the ITAM-containing cytoplasmic domains. Here we investigate the conformation and oligomeric state of several immune receptor cytoplasmic domains, using purified recombinant proteins and a variety of biophysical and biochemical techniques. We show that all of the cytoplasmic domains of ITAM-containing signaling subunits studied are oligomeric in solution, namely, T cell antigen receptor zeta, CD3epsilon, CD3delta, and CD3gamma, B cell antigen receptor Igalpha and Igbeta, and Fc receptor FcepsilonRIgamma. For zeta(cyt), the oligomerization behavior is best described by a two-step monomer-dimer-tetramer fast dynamic equilibrium with dissociation constants in the order of approximately 10 microM (monomer-dimer) and approximately 1 mM (dimer-tetramer). In contrast to the other ITAM-containing proteins, Igalpha(cyt) forms stable dimers and tetramers even below 10 microM. Circular dichroic analysis reveals the lack of stable ordered structure of the cytoplasmic domains studied, and oligomerization does not change the random-coil-like conformation observed. The random-coil nature of zeta(cyt) was also confirmed by heteronuclear NMR. Phosphorylation of zeta(cyt) and FcepsilonRIgamma(cyt) does not significantly alter their oligomerization behavior. The implications of these results for transmembrane signaling and cellular activation by immune receptors are discussed.

General numerical treatment of competitive binding kinetics: application to thrombin-dehydrothrombin-hirudin

This paper describes a general numerical method for the determination of rate constants that characterize the binding of a ligand L simultaneously and competitively to two different receptor molecules, R1 and R2. The experimental data consist of changes in the concentration of one receptor (e.g., R1) monitored over time. An example problem is represented by hirudin (L) binding to thrombin (R1) and to a chemical mutant of thrombin (R2). The published experimental data [Wedemeyer et al. (1997) Anal. Biochem. 248, 130-140], previously analyzed by using an appropriate algebraic method, were reanalyzed here by numerical integration [Kuzmic (1996) Anal. Biochem. 237, 260-273]. This general numerical method offers the following advantages. (1) It provides an estimate for the lower limit on feasible values of association rate constants. (2) It is many orders of magnitude more accurate. (3) It is easily extensible to more complicated reaction mechanisms. (4) It uses a simpler formalism and it is thus more accessible to nonmathematicians. (5) A suitable computer program for the analysis of competitive binding kinetics can be obtained via the Internet (http://www.biokin.com).