Coordination to lanthanide ions distorts binding site conformation in calmodulin.

The Ca2+-sensing protein calmodulin (CaM) is a popular model of biological ion binding since it is both experimentally tractable and essential to survival in all eukaryotic cells. CaM modulates hundreds of target proteins and is sensitive to complex patterns of Ca2+ exposure, indicating that it functions as a sophisticated dynamic transducer rather than a simple on/off switch. Many details of this transduction function are not well understood. Fourier transform infrared (FTIR) spectroscopy, ultrafast 2D infrared (2D IR) spectroscopy, and electronic structure calculations were used to probe interactions between bound metal ions (Ca2+ and several trivalent lanthanide ions) and the carboxylate groups in CaM's EF-hand ion-coordinating sites. Since Tb3+ is commonly used as a luminescent Ca2+ analog in studies of protein-ion binding, it is important to characterize distinctions between the coordination of Ca2+ and the lanthanides in CaM. Although functional assays indicate that Tb3+ fully activates many Ca2+-dependent proteins, our FTIR spectra indicate that Tb3+, La3+, and Lu3+ disrupt the bidentate coordination geometry characteristic of the CaM binding sites' strongly conserved position 12 glutamate residue. The 2D IR spectra indicate that, relative to the Ca2+-bound form, lanthanide-bound CaM exhibits greater conformational flexibility and larger structural fluctuations within its binding sites. Time-dependent 2D IR lineshapes indicate that binding sites in Ca2+-CaM occupy well-defined configurations, whereas binding sites in lanthanide-bound-CaM are more disordered. Overall, the results show that binding to lanthanide ions significantly alters the conformation and dynamics of CaM's binding sites.

Non-Additive Effects of Binding Site Mutations in Calmodulin.

Despite decades of research on ion-sensing proteins, gaps persist in the understanding of ion binding affinity and selectivity even in well-studied proteins such as calmodulin. Site-directed mutagenesis is a powerful and popular tool for addressing outstanding questions about biological ion binding and is employed to selectively deactivate binding sites and insert chromophores at advantageous positions within ion binding structures. However, even apparently nonperturbative mutations can distort the binding dynamics they are employed to measure. We use Fourier transform infrared (FTIR) and ultrafast two-dimensional infrared (2D IR) spectroscopy of the carboxylate asymmetric stretching mode in calmodulin as a mutation- and label-independent probe of the conformational perturbations induced in calmodulin's binding sites by two classes of mutation, tryptophan insertion and carboxylate side-chain deletion, commonly used to study ion binding in proteins. Our results show that these mutations not only affect ion binding but also induce changes in calmodulin's conformational landscape along coordinates not probed by vibrational spectroscopy, remaining invisible without additional perturbation of binding site structure. Comparison of FTIR line shapes with 2D IR diagonal slices provides a clear example of how nonlinear spectroscopy produces well-resolved line shapes, refining otherwise featureless spectral envelopes into more informative vibrational spectra of proteins.

Structural identifiability of equilibrium ligand-binding parameters.

Understanding the interactions of proteins with their ligands requires knowledge of molecular properties, such as binding site affinities and the effects that binding at one site exerts on binding at other sites (cooperativity). These properties cannot be measured directly and are usually estimated by fitting binding data with models that contain these quantities as parameters. In this study, we present a general method for answering the critical question of whether these parameters are identifiable (i.e., whether their estimates are accurate and unique). In cases in which parameter estimates are not unique, our analysis provides insight into the fundamental causes of nonidentifiability. This approach can thus serve as a guide for the proper design and analysis of protein-ligand binding experiments. We show that the equilibrium total binding relation can be reduced to a conserved mathematical form for all models composed solely of bimolecular association reactions and to a related, conserved form for all models composed of arbitrary combinations of binding and conformational equilibria. This canonical mathematical structure implies a universal parameterization of the binding relation that is consistent with virtually any physically reasonable binding model, for proteins with any number of binding sites. Matrix algebraic methods are used to prove that these universal parameter sets are structurally identifiable (SI; i.e., identifiable under conditions of noiseless data). A general approach for assessing and understanding the factors governing practical identifiability (i.e., the identifiability under conditions of real, noisy data) of these SI parameter sets is presented in the companion paper by Middendorf and Aldrich (2017. J. Gen. Physiol. https://doi.org/10.1085/jgp.201611703).

The structure of binding curves and practical identifiability of equilibrium ligand-binding parameters.

A critical but often overlooked question in the study of ligands binding to proteins is whether the parameters obtained from analyzing binding data are practically identifiable (PI), i.e., whether the estimates obtained from fitting models to noisy data are accurate and unique. Here we report a general approach to assess and understand binding parameter identifiability, which provides a toolkit to assist experimentalists in the design of binding studies and in the analysis of binding data. The partial fraction (PF) expansion technique is used to decompose binding curves for proteins with n ligand-binding sites exactly and uniquely into n components, each of which has the form of a one-site binding curve. The association constants of the PF component curves, being the roots of an n-th order polynomial, may be real or complex. We demonstrate a fundamental connection between binding parameter identifiability and the nature of these one-site association constants: all binding parameters are identifiable if the constants are all real and distinct; otherwise, at least some of the parameters are not identifiable. The theory is used to construct identifiability maps from which the practical identifiability of binding parameters for any two-, three-, or four-site binding curve can be assessed. Instructions for extending the method to generate identifiability maps for proteins with more than four binding sites are also given. Further analysis of the identifiability maps leads to the simple rule that the maximum number of structurally identifiable binding parameters (shown in the previous paper to be equal to n) will also be PI only if the binding curve line shape contains n resolved components.

Determination of parameter identifiability in nonlinear biophysical models: A Bayesian approach.

A major goal of biophysics is to understand the physical mechanisms of biological molecules and systems. Mechanistic models are evaluated based on their ability to explain carefully controlled experiments. By fitting models to data, biophysical parameters that cannot be measured directly can be estimated from experimentation. However, it might be the case that many different combinations of model parameters can explain the observations equally well. In these cases, the model parameters are not identifiable: the experimentation has not provided sufficient constraining power to enable unique estimation of their true values. We demonstrate that this pitfall is present even in simple biophysical models. We investigate the underlying causes of parameter non-identifiability and discuss straightforward methods for determining when parameters of simple models can be inferred accurately. However, for models of even modest complexity, more general tools are required to diagnose parameter non-identifiability. We present a method based in Bayesian inference that can be used to establish the reliability of parameter estimates, as well as yield accurate quantification of parameter confidence.