arXiv subjects

Arti Dua

Publications and source records attributed to Arti Dua.

14 recordsLinked to original sources

arXiv2024

Relaxation to statistical equilibrium in stochastic Michaelis-Menten kinetics

The equilibration of enzyme and complex concentrations in deterministic Michaelis-Menten reaction networks underlies the hyperbolic dependence between the input (substrates) and output (products). This relationship was first obtained by Michaelis and Menten and then Briggs and Haldane in two asymptotic limits: `fast equilibrium' and `steady state'. In stochastic Michaelis-Menten networks, relevant to catalysis at single-molecule and mesoscopic concentrations, the classical analysis cannot be directly applied due to molecular discreteness and fluctuations. Instead, as we show here, such networks require a more subtle asymptotic analysis based on the decomposition of the network into reversible and irreversible sub-networks and the exact solution of the chemical master equation (CME). The reversible and irreversible sub-networks reach detailed balance and stationarity, respectively, through a relaxation phase that we characterise in detail through several new statistical measures. Since stochastic enzyme kinetics encompasses the single-molecule, mesoscopic and thermodynamic limits, our work provides a broader molecular viewpoint of the classical results, in much the same manner that statistical mechanics provides a broader understanding of thermodynamics.

q-bio.MN↗

arXiv2020

Transients generate memory and break hyperbolicity in stochastic enzymatic networks

The hyperbolic dependence of catalytic rate on substrate concentration is a classical result in enzyme kinetics, quantified by the celebrated Michaelis-Menten equation. The ubiquity of this relation in diverse chemical and biological contexts has recently been rationalized by a graph-theoretic analysis of deterministic reaction networks. Experiments, however, have revealed that "molecular noise" - intrinsic stochasticity at the molecular scale - leads to significant deviations from classical results and to unexpected effects like "molecular memory", i.e., the breakdown of statistical independence between turnover events. Here we show, through a new method of analysis, that memory and non-hyperbolicity have a common source in an initial, and observably long, transient peculiar to stochastic reaction networks of multiple enzymes. Networks of single enzymes do not admit such transients. The transient yields, asymptotically, to a steady-state in which memory vanishes and hyperbolicity is recovered. We propose new statistical measures, defined in terms of turnover times, to distinguish between the transient and steady states and apply these to experimental data from a landmark experiment that first observed molecular memory in a single enzyme with multiple binding sites. Our study shows that catalysis at the molecular level with more than one enzyme always contains a non-classical regime and provides insight on how the classical limit is attained.

q-bio.MN↗

arXiv2019

Enzyme kinetics at the molecular level

The celebrated Michaelis-Menten (MM) expression provides a fundamental relation between the rate of enzyme catalysis and substrate concentration. The validity of this classical expression is, however, restricted to macroscopic amounts of enzymes and substrates and, thus, to processes with negligible fluctuations. Recent experiments have measured fluctuations in the catalytic rate to reveal that the MM equation, though valid for bulk amounts, is not obeyed at the molecular level. In this mini-review, we show how new statistical measures of fluctuations in the catalytic rate identify a regime in which the MM equation is always violated. This regime, characterized by temporal correlations between enzymatic turnovers, is absent for a single enzyme and unobservably short in the classical limit.

physics.bio-ph↗

arXiv2016

Emergence of Dynamic Cooperativity in the Stochastic Kinetics of Fluctuating Enzymes

Dynamic cooperativity in monomeric enzymes is characterized in terms of a non-Michaelis-Menten kinetic behaviour. The latter is believed to be associated with mechanisms that include multiple reaction pathways due to enzymatic conformational fluctuations. Recent advances in single-molecule fluorescence spectroscopy have provided new fundamental insights on the possible mechanisms underlying reactions catalyzed by fluctuating enzymes. Here, we present a bottom-up approach to understand enzyme turnover kinetics at physiologically relevant mesoscopic concentrations informed by mechanisms extracted from single-molecule stochastic trajectories. The stochastic approach, presented here, shows the emergence of dynamic cooperativity in terms of a slowing down of the Michaelis-Menten (MM) kinetics resulting in negative cooperativity. For fewer enzymes, dynamic cooperativity emerges due to the combined effects of enzymatic conformational fluctuations and molecular discreteness. The increase in the number of enzymes, however, suppresses the effect of enzymatic conformational fluctuations such that dynamic cooperativity emerges solely due to the discrete changes in the number of reacting species. These results confirm that the turnover kinetics of fluctuating enzyme based on the parallel-pathway MM mechanism switches over to the single-pathway MM mechanism with the increase in the number of enzymes. For large enzyme numbers, convergence to the exact MM equation occurs in the limit of very high substrate concentration as the stochastic kinetics approaches the deterministic behaviour.

physics.chem-ph↗

arXiv2015

Solvent-quality dependent contact formation dynamics in proteins

The mean time of contact formation between two ends of a protein chain shows power law dependence with respect to the number of residues, $τ_{CF} \sim N^α$. Fluorescence quenching measurements based on triplet-triplet energy transfer show variation in the value of scaling exponent $α$ for different protein-solvent systems. Here, starting from a non-Markovian diffusion equation supplemented with an exponential sink term that accounts for the energy transfer reaction between donor and acceptor groups, we calculate the mean time of contact formation using the Wilemski-Fixman closure approximation. The non-Markovian diffusion-reaction equation includes the effects of solvent quality and hydrodynamic interaction in a mean-field fashion. It shows that the contact formation dynamics is mainly governed by two time scales, the reciprocal of the intrinsic rate of quenching $(k_0^{ET})^{-1}$, and the relaxation time $τ_0 = ηb^3/k_B T$ of the coarse-grained residue of an effective size $b$ with solvent viscosity $η$. In the limit of $k_0^{ET} τ_0 \ll 1$, the dominating effect of the reaction-controlled kinetics yields the scaling exponents as $0.89$, $1.47$ and $1.79$ in poor, theta and good solvents respectively. In the opposite limit $k_0^{ET} τ_0 \gg 1$, the dominating influence of the diffusion-controlled kinetics results in $α$ as $1.90$, $2.17$, $2.36$ for a freely-draining and $1.31$, $1.77$, $2.06$ for a non-freely-draining chain in poor, theta and good solvents respectively. In the intermediate limit, $k_0^{ET} τ_0 \approx 1$, the increase in the number of residues switches the kinetics from reaction-controlled at low $N$ to diffusion-controlled at large $N$. These general results suggest that experimental estimates of the scaling exponents reflect solvent-quality dependence of the mean contact formation time in the reaction-controlled limit.

cond-mat.stat-mech↗

arXiv2015

Parallel versus off-pathway Michaelis-Menten mechanism for single-enzyme kinetics of a fluctuating enzyme

Recent fluorescence spectroscopy measurements of the turnover time distribution of single-enzyme turnover kinetics of $β$-galactosidase provide evidence of Michaelis-Menten kinetics at low substrate concentration. However, at high substrate concentrations, the dimensionless variance of the turnover time distribution shows systematic deviations from the Michaelis-Menten prediction. This difference is attributed to conformational fluctuations in both the enzyme and the enzyme-substrate complex and to the possibility of both parallel and off-pathway kinetics. Here, we use the chemical master equation to model the kinetics of a single fluctuating enzyme that can yield a product through either parallel or off-pathway mechanisms. An exact expression is obtained for the turnover time distribution from which the mean turnover time and randomness parameters are calculated. The parallel and off-pathway mechanisms yield strikingly different dependences of the mean turnover time and the randomness parameter on the substrate concentration. In the parallel mechanism, the distinct contributions of enzyme and enzyme-substrate fluctuations are clearly discerned from the variation of the randomness parameter with substrate concentration. From these general results we conclude that an off-pathway mechanism, with substantial enzyme-substrate fluctuations, is needed to rationalize the experimental findings of single-enzyme turnover kinetics of $β$-galactosidase.

physics.chem-ph↗

arXiv2014

Weak polyelectrolytes in the presence of counterion condensation with ions of variable size and polarizability

Light scattering and viscometric measurements on weak polyelectrolytes show two important aspects of counterion condensation, namely, non-monotonic variation in the polyelectrolyte size with the increase in the electrostatic strength, and, monovalent counterion selectivity in determining the nature of collapse transition at high electrostatic strengths. Here, we present a self-consistent variational theory for weak polyelectrolytes which includes the effects of the polarizability of monovalent counterions. Our theory reproduces several experimental findings including non-monotonic conformational size with the variation in the electrostatic strength and a shift from a continuous to a discontinuous collapse transition with the increase in the dipole strength of condensed ions. At low dipole strength and high electrostatic strength, our theory predicts a series of solvent quality driven size transitions spanning the re-entrant poor, theta and good solvent regimes. At high dipole strength, the size remains that of a compact globule independent of solvent quality. The dipole strength of the ion-pair formed due to counterion condensation, which depends on the size and polarizability of the monovalent counterions, is found to be an important molecular parameter in determining the nature of collapse transition, and the size of the collapsed state at high electrostatic strength.

cond-mat.stat-mech↗

arXiv2011

Non-renewal statistics in the catalytic activity of enzyme molecules at mesoscopic concentrations

Recent fluorescence spectroscopy measurements of single-enzyme kinetics have shown that enzymatic turnovers form a renewal stochastic process in which the inverse of the mean waiting time between turnovers follows the Michaelis-Menten equation. Under typical physiological conditions, however, tens to thousands of enzymes react in catalyzing thousands to millions of substrates. We study enzyme kinetics at these physiologically relevant conditions through a master equation including stochasticity and molecular discreteness. From the exact solution of the master equation we find that the waiting times are neither independent nor are they identically distributed, implying that enzymatic turnovers form a non-renewal stochastic process. The inverse of the mean waiting time shows strong departures from the Michaelis-Menten equation. The waiting times between consecutive turnovers are anti-correlated, where short intervals are more likely to be followed by long intervals and vice versa. Correlations persist beyond consecutive turnovers indicating that multi-scale fluctuations govern enzyme kinetics.

cond-mat.stat-mech↗

arXiv2009

Conformational transitions of polyelectrolytes in poor solvents

Starting with a model Hamiltonian, we study using the uniform expansion method conformational behavior of polyelectrolytes in the presence and absence of salt. The uniform expansion method yields all the important local length scales in the polyelectrolyte: the electrostatic blob size at large fraction of charges, the thermal blob size at low fraction of charges and the sizes of pearls (beads) and necklaces (strings) at intermediate fraction of charges. In the presence of salt, the electrostatic blob size depends on the ionic strength and increases with the increase in the salt concentration. We determine the salt concentration at which the pearl necklace intermediates dissolve and the nature of the transition changes from discontinuous to continuous. This critical salt concentration corresponds to the length scale where the Debye screening length is of the order of the necklace length.

cond-mat.stat-mech↗

arXiv2008

Non-Markovian fluctuations in Markovian models of protein dynamics

Recent experiments using fluorescence spectroscopy have been able to probe the dynamics of conformational fluctuations in proteins. The fluctuations are Gaussian but do not decay exponentially, and are therefore, non-Markovian. We present a theory where non-Markovian fluctuation dynamics emerges naturally from the superposition of the Markovian fluctuations of the normal modes of the protein. A Rouse-like dynamics of the normal modes provides very good agreement to the experimentally measured correlation functions. We provide simple scaling arguments rationalising our results.

cond-mat.stat-mech↗

arXiv2005

Self-consistent variational theory for globules

A self-consistent variational theory for globules based on the uniform expansion method is presented. This method, first introduced by Edwards and Singh to estimate the size of a self-avoiding chain, is restricted to a good solvent regime, where two-body repulsion leads to chain swelling. We extend the variational method to a poor solvent regime where the balance between the two-body attractive and the three-body repulsive interactions leads to contraction of the chain to form a globule. By employing the Ginzburg criterion, we recover the correct scaling for the $θ$-temperature. The introduction of the three-body interaction term in the variational scheme recovers the correct scaling for the two important length scales in the globule - its overall size $R$, and the thermal blob size $ξ_{T}$. Since these two length scales follow very different statistics - Gaussian on length scales $ξ_{T}$, and space filling on length scale $R$ - our approach extends the validity of the uniform expansion method to non-uniform contraction rendering it applicable to polymeric systems with attractive interactions. We present one such application by studying the Rayleigh instability of polyelectrolyte globules in poor solvents. At a critical fraction of charged monomers, $f_c$, along the chain backbone, we observe a clear indication of a first-order transition from a globular state at small $f$, to a stretched state at large $f$; in the intermediate regime the bistable equilibrium between these two states shows the existence of a pearl-necklace structure.

cond-mat.soft↗

arXiv2004

Chains in critical fluids and nanopores

The conformational behavior of a polymer in a critical binary solvent confined in a porous medium is studied. The size of the polymer in bulk, which is mainly governed by the correlation length of the solvent density fluctuations, depends on the proximity to the critical point of the binary mixture. We find that in contrast to the bulk behavior, the conformational properties of the polymer in a porous medium depends strongly on the pore size. The latter controls the correlation length of the solvent density fluctuations and thus determines the polymer size.

cond-mat.soft↗

arXiv2004

Constrained dynamics of a polymer ring enclosing a constant area

The dynamics of a polymer ring enclosing a constant {\sl algebraic} area is studied. The constraint of a constant area is found to couple the dynamics of the two Cartesian components of the position vector of the polymer ring through the Lagrange multiplier function which is time dependent. The time dependence of the Lagrange multiplier is evaluated in a closed form both at short and long times. At long times, the time dependence is weak, and is mainly governed by the inverse of the first mode of the area. The presence of the constraint changes the nature of the relaxation of the internal modes. The time correlation of the position vectors of the ring is found to be dominated by the first Rouse mode which does not relax even at very long times. The mean square displacement of the radius vector is found to be diffusive, which is associated with the rotational diffusion of the ring.

cond-mat.soft↗

arXiv2004

Semiflexible polymers in a random environment

We present using simple scaling arguments and one step replica symmetry breaking a theory for the localization of semiflexible polymers in a quenched random environment. In contrast to completely flexible polymers, localization of semiflexible polymers depends not only on the details of the disorder but also on the ease with which polymers can bend. The interplay of these two effects can lead to the delocalization of a localized polymer with an increase in either the disorder density or the stiffness. Our theory provides a general criterion for the delocalization of polymers with varying degrees of flexibility and allows us to propose a phase diagram for the highly folded (localized) states of semiflexible polymers as a function of the disorder strength and chain rigidity.

cond-mat.soft↗