Molecular noise-induced activator-inhibitor duality in enzyme inhibition kinetics.

Classical theories of enzyme inhibition kinetics predict a monotonic decrease in the mean catalytic activity with the increase in inhibitor concentration. The steady-state result, derived from deterministic mass action kinetics, ignores molecular noise in enzyme-inhibition mechanisms. Here, we present a stochastic generalization of enzyme inhibition kinetics to mesoscopic enzyme concentrations by systematically accounting for molecular noise in competitive and uncompetitive mechanisms of enzyme inhibition. Our work reveals an activator-inhibitor duality as a non-classical effect in the transient regime in which inhibitors tend to enhance enzymatic activity. We introduce statistical measures that quantify this counterintuitive response through the stochastic analog of the Lineweaver-Burk plot that shows a merging of the inhibitor-dependent velocity with the Michaelis-Menten velocity. The statistical measures of mean and temporal fluctuations - fractional enzyme activity and waiting time correlations - show a non-monotonic rise with the increase in inhibitors before subsiding to their baseline value. The inhibitor and substrate dependence of the fractional enzyme activity yields kinetic phase diagrams for non-classical activator-inhibitor duality. Our work links this duality to a molecular memory effect in the transient regime, arising from positive correlations between consecutive product turnover times. The vanishing of memory in the steady state recovers all the classical results.

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.

Unraveling mechanisms from waiting time distributions in single-nanoparticle catalysis.

The catalytic conversion of substrates to products at the surface of a single nanoparticle cluster can now be resolved at the molecular scale and the waiting time between individual product turnovers measured with precision. The distribution of waiting times and, in particular, their means and variances can thus be obtained experimentally. Here, we show how theoretical modeling based on the chemical master equation (CME) provides a powerful tool to extract catalytic mechanisms and rate parameters from such experimental data. Conjecturing a family of mechanisms that both include and exclude surface restructuring, we obtain the mean and variance of their waiting times from the CME. A detailed analysis of the link between mechanism topology and waiting time dispersion, then, allows us to select several candidate mechanisms, with branched topologies, that can reproduce experimental data. From these, the least complex model that best matches experimental data is chosen as the minimum model. The CME modeling extracts the Langmuir-Hinshelwood mechanism for product formation and two-pathway mechanism for product dissociation, with substantial off-pathway state fluctuations due to surface restructuring dynamics, as the minimal model consistent with data. Our work, thus, provides a mechanistic origin of the coupling between the kinetics of catalytic turnovers and surface restructuring dynamics and yields a systematic way to compute catalytic rates from distributions of waiting times between product turnovers in the presence of surface restructuring.

Emergence of dynamic cooperativity in the stochastic kinetics of fluctuating enzymes.

Dynamic co-operativity 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 co-operativity in terms of a slowing down of the Michaelis-Menten (MM) kinetics resulting in negative co-operativity. For fewer enzymes, dynamic co-operativity 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 co-operativity 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.

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 ß-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 ß-galactosidase.

Stochastic Lindemann kinetics for unimolecular gas-phase reactions.

Lindemann, almost a century ago, proposed a schematic mechanism for unimolecular gas-phase reactions. Here, we present a new semiempirical method to calculate the effective rate constant in unimolecular gas-phase kinetics through a stochastic reformulation of Lindemann kinetics. Considering the rate constants for excitation and de-excitation steps in the Lindemann mechanism as temperature dependent empirical parameters, we construct and solve a chemical master equation for unimolecular gas-phase kinetics. The effective rate constant thus obtained shows excellent agreement with experimental data in the entire concentration range in which it is reported. The extrapolated values of the effective rate constant for very low and very high concentrations of inert gas molecules are in close agreement with values obtained using the Troe semiempirical method. Stochastic Lindemann kinetics, thus, provides a simple method to construct the full falloff curves and can be used as an alternative to the Troe semiempirical method of kinetic data analysis for unimolecular gas-phase reactions.

Protein dynamics modulated electron transfer kinetics in early stage photosynthesis.

A recent experiment has probed the electron transfer kinetics in the early stage of photosynthesis in Rhodobacter sphaeroides for the reaction center of wild type and different mutants [Science 316, 747 (2007)]. By monitoring the changes in the transient absorption of the donor-acceptor pair at 280 and 930 nm, both of which show non-exponential temporal decay, the experiment has provided a strong evidence that the initial electron transfer kinetics is modulated by the dynamics of protein backbone. In this work, we present a model where the electron transfer kinetics of the donor-acceptor pair is described along the reaction coordinate associated with the distance fluctuations in a protein backbone. The stochastic evolution of the reaction coordinate is described in terms of a non-Markovian generalized Langevin equation with a memory kernel and Gaussian colored noise, both of which are completely described in terms of the microscopics of the protein normal modes. This model provides excellent fits to the transient absorption signals at 280 and 930 nm associated with protein distance fluctuations and protein dynamics modulated electron transfer reaction, respectively. In contrast to previous models, the present work explains the microscopic origins of the non-exponential decay of the transient absorption curve at 280 nm in terms of multiple time scales of relaxation of the protein normal modes. Dynamic disorder in the reaction pathway due to protein conformational fluctuations which occur on time scales slower than or comparable to the electron transfer kinetics explains the microscopic origin of the non-exponential nature of the transient absorption decay at 930 nm. The theoretical estimates for the relative driving force for five different mutants are in close agreement with the experimental estimates obtained using electrochemical measurements.

Single-molecule enzyme kinetics in the presence of inhibitors.

Recent studies in single-molecule enzyme kinetics reveal that the turnover statistics of a single enzyme is governed by the waiting time distribution that decays as mono-exponential at low substrate concentration and multi-exponential at high substrate concentration. The multi-exponentiality arises due to protein conformational fluctuations, which act on the time scale longer than or comparable to the catalytic reaction step, thereby inducing temporal fluctuations in the catalytic rate resulting in dynamic disorder. In this work, we study the turnover statistics of a single enzyme in the presence of inhibitors to show that the multi-exponentiality in the waiting time distribution can arise even when protein conformational fluctuations do not influence the catalytic rate. From the Michaelis-Menten mechanism of inhibited enzymes, we derive exact expressions for the waiting time distribution for competitive, uncompetitive, and mixed inhibitions to quantitatively show that the presence of inhibitors can induce dynamic disorder in all three modes of inhibitions resulting in temporal fluctuations in the reaction rate. In the presence of inhibitors, dynamic disorder arises due to transitions between active and inhibited states of enzymes, which occur on time scale longer than or comparable to the catalytic step. In this limit, the randomness parameter (dimensionless variance) is greater than unity indicating the presence of dynamic disorder in all three modes of inhibitions. In the opposite limit, when the time scale of the catalytic step is longer than the time scale of transitions between active and inhibited enzymatic states, the randomness parameter is unity, implying no dynamic disorder in the reaction pathway.

Nonrenewal 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. We study enzyme kinetics at physiologically relevant mesoscopic concentrations using a master equation. From the exact solution of the master equation we find that the waiting times are neither independent nor identically distributed, implying that enzymatic turnovers form a nonrenewal stochastic process. The inverse of the mean waiting time shows strong departure from the Michaelis-Menten equation. The waiting times between consecutive turnovers are anticorrelated, where short intervals are more likely to be followed by long intervals and vice versa. Correlations persist beyond consecutive turnovers indicating that multiscale fluctuations govern enzyme kinetics.

Constrained dynamics of a polymer ring enclosing a constant area.

The dynamics of an ideal polymer ring enclosing a constant 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.

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.