Kinetic NMR titration and SimFit: Tools in the Systems Chemistry Toolbox
Work in PACE lead us to the understanding of a quite bizarre NMR phenomenon, which we had seen in ours systems years before but it had escaped our attention that this is a highly valuable source of kinetic and thermodynamic information (
Figure 35
Briefly, if a usual reaction is monitored as a function of time, one observes just changes of NMR signal integrals. Growing signals are attributed to reaction products while decaying signals belong to the reaction precursors. For self-replicating systems one observes not only integral changes but also characteristic changes of the signal position, called chemical shift. The reason for this phenomenon is that such systems involve supramolecular complexes, viz. noncovalent reaction intermediates which form and decompose rapidly on the NMR time scale. Single stranded template molecules are in rapid equilibrium with hydrogen bridged complexes such as productive termolecular complexes, template duplexes, and others. An observer proton in the template senses a difference in the chemical environment, its chemical shift reflects whether the template is free in solution, has docked to its building blocks or has docked to another template molecule. If the process of sensing is slow compared to the life-time of such complexes, an average chemical shift is observed whose value depends on the concentrations of interacting molecules and thus the population structure of the system. Since the latter changes in the course of the reaction, shift-shifting reflects the process of complex repopulation. In the beginning of the reaction the shift of a template signal is close to the one of pure templates which are free in solution. Low template concentrations do not facilitate a significant involvement of complexes due to entropic reasons. At the end of the reaction we have high template concentrations but no precursors anymore. High template concentrations lead to the dimerization of templates whose NMR shift is now close to the one for the pure template duplexes. In the interim phase we have an involvement of complexes that are formed from both the template precursors and the template itself.
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". The term is in a sense, both, correct and misleading. It is correct because part of the underlaying math is similar to the one for classical NMR titrations. Also, both classical and kinetic NMR titrations are limited to exchange processes rapid enough for the NMR time scale and estimatable from Heisenberg uncertainity principle. The only difference is that the observable shift and the mole fractions are introduced as time-dependant quantities here, while they are time-invariate in classical NMR titrations. It is misleading however, because reaction-caused shift shifting is not a 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.
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). It started in the DOS environment, grew and grew over the years, became faster and faster with each computer generation, got a Windows surface while still being a script dinosaur. Nevertheless, many labs dealing with chemical self-replication including Ghadiri's, Philp's, Chmielewsk's and others routinely use it on a daily base and survive with its spartanic look and feel. 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.
In the most recent application, SimFit was employed to decipher an autocatalytic reaction network, in which the Azoarcus ribozyme catalyzes its own formation from four precursor fragments.).[54] This work is mentionable in the context of this report, because it constitutes a link between sequence information, self-generation, and metabolic autocatalysis.
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. For example, we have recently looked to the case of autocatalysis and chiral symmetry breaking in a Mannich-reaction ).[55] and found that this reaction is far away from a simple Mannich reaction. It is a whole autocatalytic network with lots of intermediates. We are currently addressing the question how 2D techniques such as COSY or DOSY can provide both, structural and colligative data for the analysis of complex systems in chemistry. "Systems NMR" may be on the doorstep soon.