A stretching force applied on a biomolecule such as a peptide or a nucleic acid strand can induce structural changes in the molecule as well as radically alter the kinetic depending on how the molecule is stretched209–211. Single molecule force spectroscopy (SMFS) setups such as AFMs, optical or magnetic tweezers employed in various modes, constant force, constant extension, force-ramp, force-jump, and more recently, extension clamp experiments212–217 enable us to study an array of systems under mechanical tension. Recent advances in single-molecule approaches have become powerful tools to provide unprecedented insights into the underlying energy landscape of proteins, probe interactions between molecules, and even ex- plore important structural transitions such as unzipping of DNA. In this chapter, we discuss how a similar idea can be exploited for rapidly constructing a detailed kinetic network model of the biomolecule using computer simulations, that is, ki- netic information pertaining to rare transitions can be recovered by stretching the biomolecule at conditions where transitions are more frequent.
5.1 Introduction One potential beneficiary of such an approach would be SMFS experiments them- selves. In such force spectroscopy (FS) experiments, typically the force applied to the system is recorded as a function of end-to-end distance, thereby producing a force-extension curve218–222. The force extension curve characterizes the molecule’s elasticity and may provide valuable new information about the protein kinetics. An applied force can alter the dynamics of the system and transitions are identified from the rips in the force-extension curves. Despite the high precision of force measure- ments, a quantitative description of the high dimensional dynamics of biomolecule under tension requires utmost care. Moreover, such conventional analysis of SMFS experiments involves an excessively coarse-grained view223–230. For instance, a 1-D reaction coordinate between the “folded” and “unfolded” states invoked in the inter- pretation of SMFS experiments hides the complexity of a rough multi-dimensional free energy landscape with multiple states and kinetic pathways231. Resolving the individual states can avoid complicated force-dependent rate maps arising from in- correctly lumping multiple intermediate states and competing pathways. Unfortu- nately, the inability to experimentally observe microscopic structural changes in a molecule presents an obstacle towards gaining higher resolution. Kinetic models derived bottom-up from molecular simulations may be able to fill gaps in our un- derstanding of the experimental force-extension curve. While this may provide un- precedented insights, it also raises broader questions whether i) detailed microscopic kinetic model can be sufficiently versatile to encompass a wide range of stretching experiments and ii) it is possible to convert between network models applicable for each type of SMFS experiment, for example, constant force or extension, without the need to generate a new model for each new experiment.
Usual analysis of SMFS experiments involves a top-down approach where the goal is to de-convolute the effects of handles, obtain the barrier height along a re- action coordinate between the “folded” and “unfolded” states of the molecule, and probe rare molecular transitions such as the dissociation of a ligand or the unfold- ing of a domain223–229. The classic Bell-Zhurkhov232 model gives the kinetic rate of rupture of a molecule, k(F), in terms of a force (F) dependent barrier height (k(F) = k0eβF x∗). Here the parameters, x∗ is the distance between the free energy minimum and the barrier along the reaction coordinate,k0 is the intrinsic rate which can be recovered from the experimental data, and β= 1/kBT, kB is the Boltzmann factor and T is the temperature. A modified model by Evans and Ritchie233 giving the distribution of forces for unfolding a molecule has been widely used to analyze data in force-ramp experiments. However, both models assume that the barrier
5.1 Introduction does not vary with the applied force. Dudko and coworkers developed a method also based on Kramer’s theory that includes a barrier that moves with an applied force to obtain an analytical expression for the force distribution for certain land- scape profiles224, namely, the cusp or linear cubic single-well free energy surfaces as function of the pulling coordinate. More recently Zhang and Dudko227 extended the formalism to treat systems with multiple barriers subjected to time dependent force to recover the force dependent rate-maps from the experimental data. The methods described above are applicable when all the intermediate states lie on a single path- way. However, when multiple states and competing pathways are involved which may have been triggered differently due to the stretching force, these will be dis- cussed with a simple example in this chapter and there exists no general formalism to describe the landscape.
Complementing the top-down approaches in silico studies have provided valu- able insights into the mechanics of molecules under tension229,234,235. Waleset al.236 applied geometry optimization techniques to study the energy landscape of two pro- teins (L and G) as a function of a static pulling force using coarse-grained models.
Methods such as steered molecular dynamics simulations205 can provide a window into the dynamics of a molecule under stretching in atomistic detail, however, finding the relevant events in folding/unfolding pathways and interpreting the calculated en- ergy barriers encountered by the molecule remains an important challenge. Markov state models (MSMs) parametrized by molecular dynamics (MD) simulations have been used widely to study folding/unfolding/dynamics of peptides in the absence of external pulling forces62. Our objective is to develop a method to predict the change in the network model at a given stretching condition (probe separation or static force) based on underlying connections between the thermodynamics and ki- netics of the system. Analogous to kinetic theories in electrochemistry (the Butler Volmer model237,238), we find that the forward and backward rates between two states depends on an intrinsic rate that quantifies the kinetic facility available for the pathway and a quantity that we term the thermodynamic disposition for the move. Upon stretching, as in a force ramp experiment, the barrier for a particular kinetic pathway can change by an amount that can be understood in terms of a simple theory if the molecule has sufficient time to visit multiple states while the average force/extension varies slowly. A time-dependent MSM based on this for- malism is the outcome of this study. Actually, the connection between MSMs at different stretching conditions via the Bell-Evans-Polanyi (BEP)207,208 principle is the cornerstone of this study. To get a detailed understanding of how the dynamics
5.2 Theoretical Basis