Kinetic NMR titration and SimFit analyses
The PACE project has contributed to two highly important techniques for studying self-replicating systems.Methods to extract information on the thermodynamic stability of supramolecular complexes from chemical shift information have been developed by Wilcox and others. They are known as NMR titrations. In a typical NMR titration one fills a series of NMR tubes with various mixtures of two interacting molecules. One concentration is usually kept constant while the other is varied. An NMR titration curve plots the observable chemical shift as a function of the concentration ratio. Fitting this curve to an equation derivable from the respective equilibrium and mass-balance equations yields the dissociation constant of the complex as well as its unknown chemical shift as the fit parameters. Repeating such titrations at various temperatures finally allows to determine the free energy and entropy of complex formation.
We have coined the means to read and understand reaction-caused shift shifting as “Kinetic NMR titration”. There is no experimenter needed to generate the various mixtures by pipetting. Instead, the system is able to produce these mixtures on its own. This in turn has the inherent advantage that kinetic titrations are in many cases much more accurate and reliable than classical NMR titrations, simply because no pipetting means no pipetting errors. We are confident that “kinetic NMR titrations” is an entry door to a whole field of applications in which advanced NMR techniques are combined with kinetic modelling to decipher complex dynamics in feedback networks.
Simultaneous fitting of integral and shift changes are carried out with SimFit, a program for nonlinear data fitting based on dynamic simulations written by GvK (1989-2008). Briefly, SimFit uses a coder/parser to translate a reaction model, viz. a set of elementary reaction equations into a set of ordinary differential equations and their Jacobian. SimFit’s coder is able to deal with fractional coefficients making it possible to deal with pseudo-equations such as needed for our square-root law of autocatalysis. Coding means setting up a number of pointer arrays so that no compilation is necessary and every encoding is done at runtime. Simfit then needs to know what the observable quantities are e.g. spectroscopic or HPLC integrals that refer to concentrations or sum or differences of concentrations of reaction species. Observables are defined and then assigned to species by means of string expressions that again set up pointer fields encoding the relationship between observables and species at run time. Kinetic data are read from external tables during run time. Numerical integration is usually done with a solver for stiff differential equations needing the Jacobian of partial derivatives of rates by concentrations. Execution time and accuracy is good because matrix operations are based on an analytical Jacobian. Nonlinear fitting is done with a Newton-Raphson algorithm or alternatively by Simplex optimisation. For the case of knowing almost nothing on rate parameters a simple stochastic optimiser is also available. Rate parameters can be fixed, variable, or coupled. Initial concentrations are also optimizable for cases where temperature equilibrium has not been settled initially. SimFit yields a multi window MDI-type output where one window holds the plots of fitted observables, residuals, fitted shifts, residuals, species concentration as a function of time, while the other contains all text based output. Text based output includes the encoded ODE system, their Jacobian, the course of optimized rate parameters, their standard errors, and covariances. In addition, SimFit allows to visualize the error-hypersurface in all N*(N-1)/2 possible twodimensional projections of parameter pairs in another graphic window.
SimFitting a whole temperature series of kinetic NMR titration data allowed for the first time the construction of the energy profile for a parabolic replicator. At the level of an experimental energy profile experimental and theoretical data derivable from QM and QD information become comparable.