[Show abstract][Hide abstract] ABSTRACT: We mobilize both a generating function approach and the theory of finite Markov processes to compute the probability of irreversible absorption of a randomly diffusing species on a lattice with competing reaction centers. We consider an N-site lattice populated by a single deep trap, and $N$-${}1$ partially absorbing traps (absorption probability $0<s<1$). The influence of competing reaction centers on the probability of reaction at a target site (the deep trap) and the mean walk length of the random walker before localization (a measure of the reaction efficiency) are computed for different geometries. Both analytic expressions and numerical results are given for reactive processes on two-dimensional surfaces of Euler characteristic $$\Omega${}=0$ and $$\Omega${}=2$. The results obtained allow a characterization of catalyst deactivation processes on planar surfaces and on catalyst pellets where only a single catalytic site remains fully active (deep trap), the other sites being only partially active as a result of surface poisoning. The central result of our study is that the predicted dependence of the reaction efficiency on system size $N$ and on $s$ is in qualitative accord with previously reported experimental results, notably catalysts exhibiting selective poisoning due to surface sites that have different affinities for chemisorption of the poisoning agent (e.g., acid zeolite catalysts). Deviations from the efficiency of a catalyst with identical sites are quantified, and we find that such deviations display a significant dependence on the topological details of the surface (for fixed values of $N$ and $s$ we find markedly different results for, say, a planar surface and for the polyhedral surface of a catalyst pellet). Our results highlight the importance of surface topology for the efficiency of catalytic conversion processes on inhomogeneous substrates, and in particular for those aimed at industrial applications. From our exact analysis we extract results for the two limiting cases $s$\approx${}1$ and $s$\approx${}0$, corresponding respectively to weak and strong catalyst poisoning (decreasing $s$ leads to a monotonic decrease in the efficiency of catalytic conversion). The results for the $s$\approx${}0$ case are relevant for the dual problem of light-energy conversion via trapping of excitations in the chlorophyll antenna network. Here, decreasing the probability of excitation trapping $s$ at sites other than the target molecule does not result in a decrease of the efficiency as in the catalyst case, but rather in enhanced efficiency of light-energy conversion, which we characterize in terms of $N$ and $s$. The one-dimensional case and its connection with a modified version of the gambler's ruin problem are discussed. Finally, generalizations of our model are described briefly.
Physical Review E 02/2015; 91(2). DOI:10.1103/PhysRevE.91.022106 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We have designed a two-dimensional, fractal-like lattice and explored, both numerically and analytically, the differences between random walks on this lattice and a regular, square-planar Euclidean lattice. We study the efficiency of diffusion-controlled processes for flows from external sites to a centrosymmetric reaction center and, conversely, for flows from a centrosymmetric source to boundary sites. In both cases, we find that analytic expressions derived for the mean walk length on the fractal-like lattice have an algebraic dependence on system size, whereas for regular Euclidean lattices the dependence can be transcendental. These expressions are compared with those derived in the continuum limit using classical diffusion theory. Our analysis and the numerical results quantify the extent to which one paradigmatic class of spatial inhomogeneities can compromise the efficiency of adatom diffusion on solid supports and of surface-assisted self-assembly in metal-organic materials.
Physical Review E 03/2014; 89(3-1):032147. DOI:10.1103/PhysRevE.89.032147 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a new approach to visualizing and quantifying the displacement of segments of Pseudomonas aeruginosa azurin in the early stages of denaturation. Our method is based on a geometrical method developed previously by the authors, and elaborated extensively for azurin. In this study, we quantify directional changes in three α-helical regions, two regions having β-strand residues, and three unstructured regions of azurin. Snapshots of these changes as the protein unfolds are displayed and described quantitatively by introducing a scaling diagnostic. In accord with molecular dynamics simulations, we show that the long α-helix in azurin (residues 54-67) is displaced from the polypeptide scaffolding and then pivots first in one direction, and then in the opposite direction as the protein continues to unfold. The two β-strand chains remain essentially intact and, except in the earliest stages, move in tandem. We show that unstructured regions 72-81 and 84-91, hinged by β-strand residues 82-83, pivot oppositely. The region comprising residues 72-91 (40 % hydrophobic and 16 % of the 128 total residues) forms an effectively stationary region that persists as the protein unfolds. This static behavior is a consequence of a dynamic balance between the competing motion of two segments, residues 72-81 and 84-91.
European Journal of Biochemistry 12/2013; 19(4-5). DOI:10.1007/s00775-013-1077-2 · 2.54 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present exact, analytic results for the mean time to trapping of a random walker on the class of deterministic Sierpinski graphs embedded in d≥2 Euclidean dimensions, when both nearest-neighbor (NN) and next-nearest-neighbor (NNN) jumps are included. Mean first-passage times are shown to be modified significantly as a consequence of the fact that NNN transitions connect fractals of two consecutive generations.
Physical Review E 11/2013; 88(5-1):052139. DOI:10.1103/PhysRevE.88.052139 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: An analytic argument is given to show that the application of the Kirkwood superposition approximation to the description of fluid correlation functions precludes the existence of a critical point. The argument holds irrespective of the dimension of the system and the specific form of the interaction potential and settles a long-standing controversy surrounding the nature of the critical behavior predicted within the approximation.
The Journal of Chemical Physics 10/2013; 139(14):141101. DOI:10.1063/1.4824388 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The analytic and numerical methods introduced previously to study the phase behavior of hard sphere fluids starting from the Yvon-Born-Green (YBG) equation under the Kirkwood superposition approximation (KSA) are adapted to the square-well fluid. We are able to show conclusively that the YBG equation under the KSA closure when applied to the square-well fluid: (i) predicts the existence of an absolute stability limit corresponding to freezing where undamped oscillations appear in the long-distance behavior of correlations, (ii) in accordance with earlier studies reveals the existence of a liquid-vapor transition by the appearance of a "near-critical region" where monotonically decaying correlations acquire very long range, although the system never loses stability.
The Journal of Chemical Physics 04/2013; 138(16):164506. DOI:10.1063/1.4801329 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We investigate the stability to structural perturbation of Pseudomonas aeruginosa azurin using a previously developed geometric model. Our analysis considers Ru(2,2',6',2″-terpyridine)(1,10-phenanthroline)(His83)-labeled wild-type azurin and five variants with mutations to Cu-ligating residues. We find that in the early stages of unfolding, the β-strands exhibit the most structural stability. The conserved residues comprising the hydrophobic core are dislocated only after nearly complete unfolding of the β-barrel. Attachment of the Ru-complex at His83 does not destabilize the protein fold, despite causing some degree of structural rearrangement. Notably, replacing the Cys112 and/or Met121 Cu ligands does not affect the conformational integrity of the protein. Notably, these results are in accord with experimental evidence, as well as molecular dynamics simulations of the denaturation of azurin.
[Show abstract][Hide abstract] ABSTRACT: Supramolecular architectures provide a reproducible template on which surficial processes can be studied. We consider the irreversible reaction A + B → C where B is a stationary reaction center and A is a coreactant diffusing on a finite, discretized d = 2 dimensional surface of a supramolecular assembly. A lattice-statistical model is developed to quantify how the reaction efficiency changes when the template is planar, Euler characteristic Ω = 0, or wrapped on the surface of a d = 3 host, Ω = 2. We find that for aperiodic or regular surfaces of finite spatial extent, dispersed Ω = 2 assemblies better optimize surficial reactions than Ω = 0 planar hosts.
Chemical Physics Letters 06/2012; 538:86–92. DOI:10.1016/j.cplett.2012.04.032 · 1.90 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We discuss here possible models for long-range electron transfer (ET) between a donor (D) and an acceptor (A) along an anharmonic (Morse–Toda) one-dimensional (1d)-lattice. First, it is shown that the electron may form bound states (solectrons) with externally, mechanically excited solitons in the lattice thus leading to one form of soliton-mediated transport. These solectrons generally move with supersonic velocity. Then, in a thermally excited lattice, it is shown that solitons can also trap electrons, forming similar solectron bound states; here, we find that ET based on hopping can be modeled as a diffusion-like process involving not just one but several solitons. It is shown that either of these two soliton-assisted modes of transport can facilitate ET over quite long distances.
International Journal of Bifurcation and Chaos 05/2012; 20(01). DOI:10.1142/S0218127410025508 · 1.08 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A comparative study of the early stages of unfolding of five proteins: cyt c, c-b562, cyt c′, azurin, and lysozyme is reported. From crystallographic data, helical regions and intervening non-helical (or ‘turning’) regions are identified in each. Exploiting a previously introduced geometrical model, the paper describes quantitatively the stepwise extension of a polypeptide chain subject to the geometrical constraint that the spatial relationship among the residues of each triplet is fixed by native-state crystallographic data. Despite differences among the above-cited proteins, remarkable universality of behavior is found in the early stages of unfolding. At the very earliest stages, internal residues in each helical region have a common unfolding history; the terminal residues, however, are extraordinarily sensitive to structural perturbations. Residues in non-helical sections of the polypeptide unfold after residues in the internal helical regions, but with increasing steric perturbation playing a dominant role in advancing denaturation.
[Show abstract][Hide abstract] ABSTRACT: We present a comparative analysis of sequential versus hierarchical mechanisms of self-assembly in supramolecular architectures. The analysis is multifaceted, drawing on and inter-relating insights from a kinetic mean-field analysis and one based on the theory of finite Markov processes, complemented by Monte Carlo calculations. We give explicit results for two reaction pathways that are likely to dominate in early stages of self-assembly, and draw attention to experimental studies to which our results pertain: crystallization of zeolites from the bulk phase and aggregation of surface-supported supramolecular structures. Among the several conclusions that can be drawn from the theory and the simulations is a crossover from one mechanism to another, depending on the values of system parameters.
[Show abstract][Hide abstract] ABSTRACT: We consider an unbiased random walk on a finite, nth generation Sierpinski gasket (or "tower") in d = 3 Euclidean dimensions, in the presence of a trap at one vertex. The mean walk length (or mean number of time steps to absorption) is given by the exact formula The generalization of this formula to the case of a tower embedded in an arbitrary number d of Euclidean dimensions is also found, and is given by This also establishes the leading large-n behavior that may be expected on general grounds, where Nn is the number of sites on the nth generation tower and is the spectral dimension of the fractal.
International Journal of Bifurcation and Chaos 11/2011; 12(11). DOI:10.1142/S0218127402006138 · 1.08 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We explore the consequences of metrically decomposing a finite phase space, modeled as
a d-dimensional lattice, into disjoint subspaces (lattices). Ergodic flows of a test particle undergoing an unbiased random walk are characterized by implementing the theory of finite Markov processes. Insights drawn from number theory are used to design the sublattices, the roles of lattice symmetry and system dimensionality are separately
considered, and new lattice invariance relations are derived to corroborate the numerical accuracy of the calculated results. We find that the reaction efficiency in a finite system
is strongly dependent not only on whether the system is compartmentalized, but also on whether the overall reaction space of the microreactor is further partitioned into separable
reactors. We find that the reaction efficiency in a finite system is strongly dependent not only on whether the system is compartmentalized, but also on whether the overall
reaction space of the microreactor is further partitioned into separable reactors. The sensitivity of kinetic processes in nanoassemblies to the dimensionality of
compartmentalized reaction spaces is quantified.
[Show abstract][Hide abstract] ABSTRACT: Quantum–mechanical studies have predicted, and experimental studies on SrCu2O3 have confirmed, that the propagation of bound pairs on even-chain copper oxide ladders is possible, but not on ladders with an odd number of legs. To study whether this quantum–mechanical lattice parity effect has a classical analog, and to document the consequences of assuming different coupling scenarios between the ladder and adjacent sublattices, we develop a classical Markovian lattice-statistical model to monitor the efficiency of migration on a composite lattice. Regions of parameter space where significant departures from results obtained via a symmetrical random walk are identified.
Chemical Physics Letters 09/2011; 514(1):88-93. DOI:10.1016/j.cplett.2011.08.019 · 1.90 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We use an analytic criterion for vanishing of exponential damping of
correlations developed previously (Piasecki et al, J. Chem. Phys., 133, 164507,
2010) to determine the threshold volume fractions for structural transitions in
hard sphere systems in dimensions D=3,4,5 and 6, proceeding from the YBG
hierarchy and using the Kirkwood superposition approximation. We conclude that
the theory does predict phase transitions in qualitative agreement with
numerical studies. We also derive, within the superposition approximation, the
asymptotic form of the analytic condition for occurence of a structural
transition in the D->Infinity limit .
The Journal of Chemical Physics 08/2011; 135(8):084509. DOI:10.1063/1.3622597 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A geometrical model has been developed to describe the early stages of unfolding of cytochromes c′ and c-b562. Calculations are based on a step-wise extension of the polypeptide chain subject to the constraint that the spatial relationship among the residues of each triplet is fixed by the native-state crystallographic data. The response of each protein to these structural perturbations allows the evolution of each of the four helices in these two proteins to be differentiated. It is found that the two external helices in c′ unfold before its two internal helices, whereas exactly the opposite behaviour is demonstrated by c-b562. Each of these cytochromes has an extended, internal, non-helical (‘turning’) region that initially lags behind the most labile helix but then, at a certain stage (identified for each cytochrome), unravels before any of the four helices present in the native structure. It is believed that these predictions will be useful in guiding future experimental studies on the unfolding of these two cytochromes.
[Show abstract][Hide abstract] ABSTRACT: We study the early stages of self-assembly of elementary building blocks of nanophase materials, considering explicitly their structure and the symmetry and the dimensionality of the reaction space. Previous work [Kozak et al., J. Chem. Phys. 134, 154701 (2007)] focused on characterizing self-assembly on small square-planar templates. Here we consider larger lattices of square-planar symmetry having N = 255 sites, and both hexagonal and triangular lattices of N = 256 sites. Furthermore, to assess the consequences of a depletion zone above a basal layer (λ = 1), we study self-assembly on an augmented diffusion space defined by λ = 2 and λ = 5 stacked layers having the same characteristics as the basal plane. The effective decrease in the efficiency of self-assembly of individual nanophase units when the diffusion space is expanded, by increasing the template size and/or by enlarging the depletion zone, is then quantified. The results obtained reinforce our earlier conclusion that the most significant factor influencing the kinetics of formation of a final self-assembled unit is the number of reaction pathways from one or more precursor states. We draw attention to the relevance of these results to zeolite synthesis and reactions within pillared clays.
The Journal of Chemical Physics 02/2011; 134(6):064701. DOI:10.1063/1.3541822 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We have developed a geometrical model to study the unfolding of iso-1 cytochrome c. The model draws on the crystallographic data reported for this protein. These data were used to calculate the distance between specific residues in the folded state, and in a sequence of extended states defined by n= 3, 5, 7, 9, 11, 13, and 15 residue units. Exact calculations carried out for each of the 103 residues in the polypeptide chain demonstrate that different regions of the chain have different unfolding histories. Regions where there is a persistence of compact structures can be identified, and this geometrical characterization is fully consistent with analyses of time-resolved fluorescence energy-transfer (TrFET) data using dansyl-derivatized cysteine side-chain probes at positions 39, 50, 66, 85, and 99. Our calculations were carried out assuming that different regions of the polypeptide chain unfold synchronously. To test this assumption, we performed lattice Monte Carlo simulations to study systematically the possible importance of asynchronicity. Our calculations show that small departures from synchronous dynamics can arise if displacements of residues in the main body of the chain are much more sluggish than near-terminal residues.