Proteins under native state conditions exhibit a wide range of motions enabling the performance of their biological function. Fast protein motions (pico-nanosecond timescale) are local and involve conformational uctuations of a few angstroms, including side chain rotations and small backbone

Introduction 127 Computational Approaches to Describe Slow Conformational Dynamics 132

ENM 132 Slow Dynamic Fingerprints of the Amino Acid Kinase Family: NAGK Dynamic Paradigm 133 Catalytic Prociency of Di erent Conformations 135 Conformational Compression in the Chemical Step 136 Is Enzyme Dynamics the Rate-Limiting Step? 138

Role of Oligomeric Assembly in Functional Motions 139 Hexameric NAGK 139 Hexameric UMP Kinase 141

Motions in Rhamnulose-1-Phosphate and Fuculose-1-Phosphate Aldolases 141 Concluding Remarks 144 Acknowledgments 145 References 145

movements. By contrast, at slow timescales (micro-milliseconds and beyond), large collective motions of protein domains and entire subunits in oligomeric assemblies allow changes in conformation of tenths of angstroms that are typically associated to active site opening/closing in ligand binding and vast structural rearrangements in allosteric events. It is the balanced interplay of this hierarchy of motions that allows proteins to adopt conformations outstandingly complementary to their binding partners. Toward unveiling the relationship between protein function and dynamics, computational methods, such as those covered in this book, in combination with biophysics experiments have been successful in describing relevant dynamic properties, but still many fundamental questions remain to be answered. For instance, why should enzymes move? Of course, they move because temperature, at the molecular level, shakes everything. But evolution can regulate the rigidity of proteins in compliance with stability, as observed in thermophilicmesophilic pairs of enzymes (Kumar and Nussinov 2001; Razvi and Scholtz 2006; Sterpone and Melchionna 2012; Vieille and Zeikus 2001; Wolf-Watz et al. 2004). So it seems that enzymes could be much more rigid than what they are; suggesting that exibility is related to their function and likely to their capacity to evolve toward new functions (Dellus-Gur et al. 2013). While there are many experimental (Bhabha et al. 2011; Cameron and Benkovic 1997; Eisenmesser et al. 2005; Henzler-Wildman et al. 2007a,b; Rajagopalan et al. 2002; Wolf-Watz et al. 2004) and computational (Agarwal 2005, 2006; Agarwal et al. 2002, 2004; Hammes-Schi er and Benkovic 2006; Ma et al. 2000; Yang and Bahar 2005) results that conrm a linkage between enzyme exibility and function, some computational studies indicate the e ects of dynamics are, if any, too small (García-Meseguer et al. 2013; Glowacki et al. 2012; Kamerlin and Warshel 2010a,b; Martí et al. 2003; Olsson et al. 2006; Pisliakov et al. 2009). Indeed, there are di erent conceptions of what “dynamical e ects” are and this has been the core of controversy in recent years. On one hand there are arguments based on transition state theory (TST). is is an equilibrium theory based on stationary properties, and therefore deviations from this theory are called dynamical e ects. ese deviations are di erent in the enzyme and in water, but their e ect on catalysis is rather small (García-Meseguer et al. 2013; Kamerlin and Warshel 2010a,b; Kanaan et al. 2010; Olsson et al. 2006; Roca et al. 2010). On the other hand, many experimentalists agree that there is ample evidence that the dynamics of enzymes is a prerequisite for their function and that altering the dynamics a ects their rates (Agarwal et al. 2002; Benkovic and Hammes-Schi er 2003; Bhabha et al. 2011; Eisenmesser et al. 2002, 2005; Engelkamp et al. 2006; English et al. 2006; Hammes-Schi er and Benkovic 2006; Min et al. 2005; Osborne et al. 2001; Rajagopalan et al. 2002). is, of course, is not in contradiction with an explanation based on TST, which would describe these e ects of dynamics as nondynamical e ects! As

long as the chemical step remains the rate-limiting step, TST remains valid. But even when that step is not rate-limiting, TST could be applied if one can dene the transition state for a protein motion. On the basis of the large number of computational studies that reproduced enzyme rate constants by only modeling the chemical step with hybrid quantum mechanics/molecular mechanics (QM/MM) methods, this step seems to be rate-limiting for several enzymes, but that also remains controversial (Benkovic and HammesSchi er 2003; Bhabha et al. 2011; Cameron and Benkovic 1997; Eisenmesser et al. 2002, 2005; Hammes-Schi er and Benkovic 2006; Wolf-Watz et al. 2004). Natural selection only puts evolutionary pressure to the rate-limiting step: once the chemical step is no longer the rate-limiting step, it will not decrease much further, and thus will remain relatively close the new rate-limiting step (e.g., large-scale enzyme motions). is turns the identication and modeling of the rate-limiting step into a computational challenge. We believe part of the seemingly contradictory facts in this eld arise from this evolutionary conundrum.