Lash Miller Chemical Laboratories

80 St. George St.

University of Toronto

Toronto, Ontario, Canada M5S 1A1

**Phone:** (416)-978-4376
**Fax:** (416)-978-8775
**Office:** Lash Miller, Room 239
**E-mail:** jmschofi@chem.utoronto.ca

The first project focuses mainly on the role of charge and aquation state in the determination of macromolecular structure in biological systems. There has been very little theoretical work on how the conformation of an isolated macromolecule depends on the number of excess charges bound to it, in spite of the fact that many protein and polypeptide molecules contain ions and are present in a variety of charge states at biological pH conditions. We are working on new molecular dynamics and Monte-Carlo simulation methods which are designed to reproduce the correct statics and dynamics of charged polypeptides and other molecules in which proton transfer is important. The goal of the work is to probe how the transitions between folded and extended conformers depend on temperature, aqueous solvation and charge state of the molecule.

Another project being pursues concerns the folding dynamics of simple models of protein systems. In protein chains, the competition between energy (or enthalpy) and entropy (energy density of conformational states) leads to transitions between denatured, non-native configurations and compact folded structures that play a specific function in biological systems. These competing interactions can also give rise to a wealth of other structural transitions among meta-stable states. This project is concerned with understanding the factors that influence the rate of conformational transitions and how the temperature behavior of the rates and the folding profile relate to features of the free energy surface, such as funnels.

The use of a discontinuous model of residue interactions has characteristics that may be exploited to probe the connection between the free energy surface and the folding dynamics. The interactions, tailored to produce secondary structural elements observed in real proteins such as alpha-helices and beta-sheets, may be used to unambiguously define conformations of the protein and permit temperature-independent entropic differences between states to be computed thereby allowing the free energies of all states to be computed at any temperature. Furthermore, if monomers in the protein-like chain experience rapid collisions with an effective solvent that leads to rapid decay of bead velocity autocorrelation functions, monomer dynamics may be described by the Smoluchowski equation. From the Smoluchowski equation, projection operator methods can be used to define expressions for transition rates between configurations in a Markov limit where bead correlations are short-lived on the time scale of transitions between states. Under conditions in which bead correlations are short-lived on the time scale of transitions between states, rate constants can be calculated from spectral analysis of a projected Smoluchowski evolution operator in terms of temperature-independent, one-dimensional integrals of probability densities for bonding distances that can be constructed using analytical fits to estimated cumulative distribution functions. The cumulative distribution functions and relative entropies for bonding can be efficiently computed using parallel tempering algorithms adapted to preferentially sample connected states that differ by a single bond to minimize statistical uncertainties of the computed rates. The forward and reverse rate constants for transitions between states can be expressed as a temperature-dependent linear combination of two effective rates that are temperature independent.

Although the resulting system of equations is equivalent to Markov models of proteins with continuous interaction potentials, here the Markov model is actually constructed from a physical system. Furthermore, unlike in standard systems, the discontinuous interaction potentials enable configurations to be identified without resorting to carrying out simulations and allow the computation of rate constants at any temperature. The resulting linear system can be readily solved to characterize the relaxation profile of an ensemble of unfolded configurations to the folded state as a function of temperature. The multi-exponential nature of the profile can be characterized by looking at the relative weights of eigenvectors of the linear system to distinguish between stretched and compressed-exponential kinetics.

The Schofield group is working towards developing discontinuous potential models of protein-like chains by treating some of the system, such as alpha-helices, as rigid objects that are linked together by flexible regions of the system. The interactions of residues and fragments can be constructed using discretizations of continuous potential interactions, while the dynamics of the rigid fragments can simulated using rigid body discontinuous molecular dynamics. By eliminating small-scale motions of rigid components of the biomolecule, one can focus on important motions of domains of the protein system at long times, such as hinge-bending, partial-refolding or shear domain motion.

Specific questions that will be addressed in this project include: How does the overall complexity of the connectivity map of available states influence the multi-exponential relaxation to the equilibrium population? What is the range of validity of the expressions for the rate constants? When are non-Markovian effects important? Do what extent are motions in the folding process of proteins diffusive?

Our beowulf cluster racaille: Photos and description

Positions available in the Schofield group

Polypeptide Structure

Proton transfer dynamics

Nonadiabatic MD

Stochastic models of complex systems

Dynamics in glassy systems

Rigid body dynamics

Chem1485: Dynamics in Liquids

Chem1464: Foundations of Molecular Simulation

CHM427 and CHM1480: Statistical Mechanics

**J. Schofield**and H. Bayat, "Derivation of a Markov state model of the dynamics of a protein-like chain immersed in an implicit solvent" ,*J. Chem. Phys.***141**, 095101 (2014) (19 pages)-
S. Jalili, E. Hosseinzadeh and

**J. Schofield,**"Study of atomic and molecular oxygen chemisorption on $BC_3$ nanotubes with Stone–Wales defects using density functional theory",*Chem. Phys.***438**, 16-22 (2014) -
C.-Yu Hsieh,

**J. Schofield,**and R. Kapral, Forward-Backward solution of quantum-classical Liouville equation in the adiabatic mapping basis,*Mol. Phys.***111**, 3546-3554 (2013) -
S. Jalili, L. Karami and

**J. Schofield,**Study of base pair mutations in proline-rich homeodomain (PRH)–DNA complexes using molecular dynamics,*Eur. Biophys. J.***42**, (2013) -
S. Jalili, M. Akhavan and

**J. Schofield,**Study of titanium adsorption on perfect and defected BC3 nanotubes using density function theory",*Mol. Phys.***111**, (2013) -
R. Raghu and

**J. Schofield,**"Simulation of tethered oligomers in nanochannels using multi-particle collision dynamics",*J. Chem. Phys.***137**, 014901 (2012) -
S. Jalili, M. Akhavan and

**J. Schofield,**"Electronic and structural properties of BC3 nanotubes with defects ",*J. Phys. Chem. C***116**, 13225-13230 (2012) H. Bayat Movahed, R. van Zon and

**J. Schofield**, "Free energy landscape of protein-like chains with discontinuous potentials" ,*J. Chem. Phys.***136**, 245103 (2012) (13 pages)**J. Schofield,**P. Inder, and R. Kapral, "Modeling of solvent flow effects in enzyme catalysis under physiological conditions" ,*J. Chem. Phys.***136**, 205101 (2012), (14 pages)-
S. Jalili, C. Mochani, M. Akhavan and

**J. Schofield,**"Molecular dynamics simulation of a graphite-supported copper nanocluster: thermodynamic properties and gas adsorption",*Mol. Phys.***110**, 267-276 (2012). A. Kelly, R. van Zon,

**J. Schofield,**and R. Kapral, "Mapping quantum-classical Liouville equation: Projectors and trajectories" ,*J. Chem. Phys.***136**, 084101 (2012), (14 pages)-
R. Raghu and

**J. Schofield,**"Simulation of Pressure-Driven Flows in Nanochannels Using Multiparticle Collision Dynamics",*J. Phys. Chem. C***114**, 20659 - 20671 (2010) R. van Zon and

**J. Schofield,**"Constructing smooth potentials of mean force, radial distribution functions, and probability densities from sampled data" ,*J. Chem. Phys.***132**, 154110 (2010), (13 pages)R. van Zon, L. Hernandez de la Pena, G. Peslherbe and

**J. Schofield,**"Quantum free energy differences from non-equilibrium path integrals. I. Methods and numerical application" ,*Phys. Rev. E***78**, 041103 (2008), (11 pages)R. van Zon, L. Hernandez de la Pena, G. Peslherbe and

**J. Schofield,**"Quantum free energy differences from non-equilibrium path integrals. II. Convergence properties for the harmonic oscillator" ,*Phys. Rev. E***78**, 041104 (2008), (14 pages)R. van Zon and

**J. Schofield,**"Event-driven dynamics of rigid bodies interacting via discretized potentials" ,*J. Chem. Phys.***128**, 154119 (2008), (9 pages)R. van Zon, I.P. Omelyan and

**J. Schofield,**"Efficient algorithms for rigid body integration using optimized splitting methods and exact free rotational motion" ,*J. Chem. Phys.***128**, 136102 (2008), (2 pages)R. van Zon and

**J. Schofield,**"Symplectic algorithms for simulations of rigid body systems using the exact solution of free motion" ,*Phys. Rev. E***75**, 056701 (2007), (5 pages)-
R. van Zon and

**J. Schofield,**"Numerical implementation of the exact dynamics of free rigid bodies" ,*J. Comp. Phys.***225**, 145-164 (2007) -
L. Hernandez de la Pena, R. van Zon,

**J. Schofield**and S. Opps, "Discontinuous molecular dynamics for semi-flexible and rigid bodies" ,*J. Chem. Phys.***126**, 074105 (2007), (13 pages) -
L. Hernandez de la Pena, R. van Zon,

**J. Schofield**and S. Opps, "Discontinuous molecular dynamics for rigid bodies: Applications" ,*J. Chem. Phys.***126**, 074106 (2007), (12 pages) -
**J. Schofield,**"Quantum effects in ab-initio calculations of rate constants for chemical reactions occuring in the condensed phase" ,*Theor. Chem. Acc.***116**, 18-30 (2006) -
R. van Zon and

**J. Schofield,**"Mode-Coupling Theory for Multiple-Time Correlation Functions of Tagged Particle Densities and Dynamical Filters Designed for Glassy Systems" ,*J. Phys. Chem. B***109**, 21425 - 21436 (2005) -
R. van Zon and

**J. Schofield,**"Glassy dynamics and domains: Explicit results for the East model" ,*J. Chem. Phys.***122**, 194502 (2005), (15 pages) -
A. Plyukhin and

**J. Schofield,**"Langevin equation for the extended Rayleigh model with an asymmetric bath",*Phys. Rev. E***69**, 021112 (2004), (7 pages) -
A. Plyukhin and

**J. Schofield,**"On the Langevin equation for the Rayleigh model with finite-ranged interactions",*Phys. Rev. E***68**, 041107 (2003), (15 pages) -
R. Iftimie, D. Salahub and

**J. Schofield,**"An efficient Monte Carlo method for calculating ab initio transition state theory reaction rates in solution" ,*J. Chem. Phys.***119**, 11285 - 11297 (2003) -
R. Iftimie and

**J. Schofield,**"The separation of quantum and classical behavior in proton transfer reactions: Implications from studies of secondary kinetic isotope effects",*Int. Journal of Quantum Chemistry***91**, 404-413 (2003) -
A. Plyukhin and

**J. Schofield,**"Trapping, reflection and fragmentation in a classical model of atom-lattice collisions",*Phys. Rev. E***65**, 026603 (2002), (8 pages) -
S. Opps and

**J. Schofield,**"Monte Carlo methods designed for parallel computation",*To appear in the conference proceedings of High Performance Computer Systems and Applications, Kluwer Academic Press*, 2002 -
C. C. Wan and

**J. Schofield,**"Solutions of mixed quantum-classical dynamics in multiple dimensions using classical trajectories",*J. Chem. Phys.***116**, 494-506 (2002) -
R. van Zon and

**J. Schofield,**"A mode-coupling theory of multi-point and multi-time correlation functions",*Phys. Rev. E***65**, 01106 (2002) (17 pages) -
R. van Zon and

**J. Schofield,**"Multiple-point and multiple-time correlation functions in a hard sphere liquid" ,*Phys. Rev. E***65**, 01107 (2002) (12 pages) -
A. Plyukhin and

**J. Schofield,**"Stochastic dynamics with a mesoscopic bath",*Phys. Rev. E***64**, (2001) 041103 (10 pages) -
R. Iftimie and

**J. Schofield,**"Reaction mechanism and isotope effects derived from centroid transition state theory in intra-molecular proton transfer reactions",*J. Chem. Phys.***115**, (2001) 5891-5902 -
A. Plyukhin and

**J. Schofield,**"A stochastic model related to the Klein-Gordon equation",*Phys. Rev. E***64**, (2001) 037101 (3 pages) -
S. Opps and

**J. Schofield,**"Extended state space Monte-Carlo methods",*Phys. Rev. E***63**, (2001) 56701-56712 -
R. Iftimie and

**J. Schofield,**"Efficient ab initio sampling methods in rate constant calculations for proton-transfer reactions"*J. Chem. Phys.***114**, (2001) 6763-6773 -
R. Iftimie, D. Salahub, D. Wei and

**J. Schofield,**"Using a classical potential as an efficient importance function for sampling an ab-initio potential",*J. Chem. Phys.***113,**(2000) 4852-4862 -
C. C. Wan and

**J. Schofield,**"Mixed quantum-classical molecular dynamics: Aspects of the multithreads algorithm",*J. Chem. Phys.***113,**(2000) 7047-7054 -
C. C. Wan and

**J. Schofield,**"Exact and asymptotic solutions of the mixed quantum-classical Liouville equation",*J. Chem. Phys.***112,**(2000) 4447-4459 -
A. H. Marcus,

**J. Schofield**and S. A. Rice, "Experimental Observations of Non-Gaussian Behavior and Stringlike Cooperative Dynamics in Concentrated Quasi-Two-Dimensional Colloidal Liquids",*Phys. Rev. E.***60**,5725-5736, 1999 -
**J. Schofield**and M. Ratner, "Monte-Carlo Methods For Short Polypeptides",*J. Chem. Phys.***109,**(1998) 9177-9191 -
**J. Schofield**, A.H. Marcus and S.A. Rice, "The Dynamics of Quasi Two Dimensional Colloidal Suspensions",*J. Phys. Chem*.**100**, (1996) 18950-18961. -
**J. Schofield**and S.A. Rice, "Backbone Ordering in Amphiphile Monolayers",*J. Chem. Phys*.**103**, (1995) 5792-5801. -
**J. Schofield**and I. Oppenheim, "Mode Coupling in Nonequilibrium Granular Flow Systems",,*Physica***A 204**, (1994) 555-605 -
**J. Schofield**and I. Oppenheim, "The hydrodynamics of inelastic granular systems"*Physica***A 196**, (1993) 209-240 -
**J. Schofield**and I. Oppenheim, "Mode Coupling and tagged particle correlation functions: the Stokes-Einstein law",*Physica***A 187**, (1992) 210-242 -
**J. Schofield**and I. Oppenheim, "Mode Coupling and generalized hydrodynamics",*Physica***A 181**, (1992) 89-135

Class notes for ENV235

Class notes for CHM427/1480

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Created September 15, 1997. Last updated September, 2014.