Advanced path-sampling methods exploit reduced physical models or biasing to produce plausible transitions while balancing accuracy and efficiency, but quantifying their accuracy relative to other numerical and experimental data has been challenging. Indeed, new horizons in elucidating protein function necessitate that present methodologies be revised to more seamlessly and quantitatively integrate a spectrum of methods, both numerical and experimental. In this dissertation, experimental and computational methods are put into perspective using the enzyme adenylate kinase (AdK) as an illustrative example. We introduce Path Similarity Analysis (PSA)—an integrative computational framework developed to quantify transition path similarity. PSA not only reliably distinguished AdK transitions by the originating method, but also traced pathway differences between two methods back to charge-charge interactions (neglected by the stereochemical model, but not the all-atom force field) in several conserved salt bridges. Cryo-electron microscopy maps of the transporter Bor1p are directly incorporated into EqMD simulations using MD flexible fitting to produce viable structural models and infer a plausible transport mechanism. Conforming to the theme of integration, a short compendium of an exploratory project—developing a hybrid atomistic-continuum method—is presented, including initial results and a novel fluctuating hydrodynamics model and corresponding numerical code.
representation theory of the Lorentz and Poincare groups, and a review of some basic rela- ´
tivistic wave equations that will play an important role in the work to follow. In Chapter 2,
a complex covariant form of the classical Maxwell’s equations in a moving medium or at
rest is introduced. In addition, a compact, Lorentz invariant, form of the energy-momentum
tensor is derived. In chapter 3, the concept of photon helicity is critically analyzed and its
connection with the Pauli-Lubanski vector from the viewpoint of the complex electromag- ´
netic field, E+ iH. To this end, a complex covariant form of Maxwell’s equations is used.
Chapter 4 analyzes basic relativistic wave equations for the classical fields, such as Dirac’s
equation, Weyl’s two-component equation for massless neutrinos and the Proca, Maxwell
and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir ´
operators of the Poincare group. A connection between the spin of a particle/field and ´
consistency of the corresponding overdetermined system is emphasized in the massless
case. Chapter 5 focuses on the so-called generalized quantum harmonic oscillator, which
is a Schrodinger equation with a time-varying quadratic Hamiltonian operator. The time ¨
evolution of exact wave functions of the generalized harmonic oscillators is determined
in terms of the solutions of certain Ermakov and Riccati-type systems. In addition, it is
shown that the classical Arnold transform is naturally connected with Ehrenfest’s theorem
for generalized harmonic oscillators. In Chapter 6, as an example of the usefulness of the
methods introduced in Chapter 5 a model for the quantization of an electromagnetic field
in a variable media is analyzed. The concept of quantization of an electromagnetic field
in factorizable media is discussed via the Caldirola-Kanai Hamiltonian. A single mode
of radiation for this model is used to find time-dependent photon amplitudes in relation
to Fock states. A multi-parameter family of the squeezed states, photon statistics, and the
uncertainty relation, are explicitly given in terms of the Ermakov-type system.
Temperature and polarizability effects on electron transfer in biology and artificial photosynthesis
(ET) and then provide modifications to the model. A confirmation of the inverted energy
gap law, which is the cleanest verification so far, is presented for donor-acceptor complexes.
In addition to the macroscopic properties of the solvent, the physical properties of the solvent
are incorporated in the model via the microscopic solvation model. For the molecules
studied in this dissertation, the rate constant first increases with cooling, in contrast to the
prediction of the Arrhenius law, and then decreases at lower temperatures. Additionally,
the polarizability of solute, which was not considered in the original Marcus theory, is included
by the Q-model of ET. Through accounting for the polarizability of the reactants, the
Q-model offers an important design principle for achieving high performance solar energy
conversion materials. By means of the analytical Q-model of ET, it is shown that including
molecular polarizability of C60 affects the reorganization energy and the activation barrier
of ET reaction.
The theory and Electrochemistry of Ferredoxin and Cytochrome c are also investigated.
By providing a new formulation for reaction reorganization energy, a long-standing disconnect
between the results of atomistic simulations and cyclic voltametery experiments is
resolved. The significant role of polarizability of enzymes in reducing the activation energy
of ET is discussed. The binding/unbinding of waters to the active site of Ferredoxin leads
to non-Gaussian statistics of energy gap and result in a smaller activation energy of ET.
Furthermore, the dielectric constant of water at the interface of neutral and charged
C60 is studied. The dielectric constant is found to be in the range of 10 to 22 which is
remarkably smaller compared to bulk water( 80). Moreover, the interfacial structural
crossover and hydration thermodynamic of charged C60 in water is studied. Increasing the
charge of the C60 molecule result in a dramatic structural transition in the hydration shell,
which lead to increase in the population of dangling O-H bonds at the interface.
Several experimental measurements can probe diffusion coefficients at the single-molecule and bulk level. The target of this thesis is on single-molecule methods, which can assess diffusion coefficients at the individual molecular level. For instance, super resolution methods like stochastic optical reconstruction microscopy (STORM) and photo activated localization microscopy (PALM), have a high spatial resolution with the cost of lower temporal resolution. Also, there is a different group of methods, such as MINFLUX, multi-detector tracking, which can track a single molecule with high spatio-temporal resolution. The problem with these methods is that they are only applicable to very diluted samples since they need to ensure existence of a single molecule in the region of interest (ROI).
In this thesis, the goal is to have the best of both worlds by achieving high spatio-temporal resolutions without being limited to a few molecules. To do so, one needs to refocus on fluorescence correlation spectroscopy (FCS) as a method that applies to both in vivo and in vitro systems with a high temporal resolution and relies on multiple molecules traversing a confocal volume for an extended period of time. The difficulty here is that the interpretation of the signal leads to different estimates for the kinetic parameters such as diffusion coefficients based on a different number of molecules we consider in the model. It is for this reason that the focus of this thesis is now on using Bayesian nonparametrics (BNPs) as a way to solve this model selection problem and extract kinetic parameters such as diffusion coefficients at the single-molecule level from a few photons, and thus with the highest temporal resolution as possible.
Enzymes in biology’s energy chains operate with low energy input distributed through multiple electron transfer steps between protein active sites. The general challenge of biological design is how to lower the activation barrier without sacrificing a large negative reaction free energy. We show that this goal is achieved through a large polarizability of the active site. It is polarized by allowing a large number of excited states, which are populated quantum mechanically by electrostatic fluctuations of the protein and hydration water shells. This perspective is achieved by extensive mixed quantum mechanical/molecular dynamics simulations of the half reaction of reduction of cytochrome c. The barrier for electron transfer is consistently lowered by increasing the number of excited states included in the Hamiltonian of the active site diagonalized along the classical trajectory. We suggest that molecular polarizability, in addition to much studied electrostatics of permanent charges, is a key parameter to consider in order to understand how enzymes work.