411 research outputs found
Rate constants for proteins binding to substrates with multiple binding sites using a generalized forward flux sampling expression
To predict the response of a biochemical system, knowledge of the intrinsic and effective rate constants of proteins is crucial. The experimentally accessible effective rate constant for association can be decomposed in a diffusion-limited rate at which proteins come into contact and an intrinsic association rate at which the proteins in contact truly bind. Reversely, when dissociating, bound proteins first separate into a contact pair with an intrinsic dissociation rate, before moving away by diffusion. While microscopic expressions exist that enable the calculation of the intrinsic and effective rate constants by conducting a single rare event simulation of the protein dissociation reaction, these expressions are only valid when the substrate has just one binding site. If the substrate has multiple binding sites, a bound enzyme can, besides dissociating into the bulk, also hop to another binding site. Calculating transition rate constants between multiple states with forward flux sampling requires a generalized rate expression. We present this expression here and use it to derive explicit expressions for all intrinsic and effective rate constants involving binding to multiple states, including rebinding. We illustrate our approach by computing the intrinsic and effective association, dissociation, and hopping rate constants for a system in which a patchy particle model enzyme binds to a substrate with two binding sites. We find that these rate constants increase as a function of the rotational diffusion constant of the particles. The hopping rate constant decreases as a function of the distance between the binding sites. Finally, we find that blocking one of the binding sites enhances both association and dissociation rate constants. Our approach and results are important for understanding and modeling association reactions in enzyme-substrate systems and other patchy particle systems and open the way for large multiscale simulations of such systems
The magnitude of the intrinsic rate constant: How deep can association reactions be in the diffusion limited regime?
Intrinsic and effective rate constants have an important role in the theory of diffusion-limited reactions. In a previous paper, we provide detailed microscopic expressions for these intrinsic rates [A. Vijaykumar, P. G. Bolhuis, and P. R. ten Wolde, Faraday Discuss. 195, 421 (2016)], which are usually considered as abstract quantities and assumed to be implicitly known. Using these microscopic expressions, we investigate how the rate of association depends on the strength and the range of the isotropic potential and the strength of the non- specific attraction in case of the anisotropic potential. In addition, we determine the location of the interface where these expressions become valid for anisotropic potentials. In particular, by investigating the particles' orientational distributions, we verify whether the interface at which these distributions become isotropic agrees with the interface predicted by the effective association rate constant. Finally, we discuss how large the intrinsic association rate can become, and what are the consequences for the existence of the diffusion limited regime. Published by AIP Publishing
When it Pays to Rush: Interpreting Morphogen Gradients Prior to Steady-State
During development, morphogen gradients precisely determine the position of
gene expression boundaries despite the inevitable presence of fluctuations.
Recent experiments suggest that some morphogen gradients may be interpreted
prior to reaching steady-state. Theoretical work has predicted that such
systems will be more robust to embryo-to-embryo fluctuations. By analysing two
experimentally motivated models of morphogen gradient formation, we investigate
the positional precision of gene expression boundaries determined by
pre-steady-state morphogen gradients in the presence of embryo-to-embryo
fluctuations, internal biochemical noise and variations in the timing of
morphogen measurement. Morphogens that are direct transcription factors are
found to be particularly sensitive to internal noise when interpreted prior to
steady-state, disadvantaging early measurement, even in the presence of large
embryo-to-embryo fluctuations. Morphogens interpreted by cell-surface receptors
can be measured prior to steady-state without significant decrease in
positional precision provided fluctuations in the timing of measurement are
small. Applying our results to experiment, we predict that Bicoid, a
transcription factor morphogen in Drosophila, is unlikely to be interpreted
prior to reaching steady-state. We also predict that Activin in Xenopus and
Nodal in zebrafish, morphogens interpreted by cell-surface receptors, can be
decoded in pre-steady-state.Comment: 18 pages, 3 figure
Radial Squeezed States and Rydberg Wave Packets
We outline an analytical framework for the treatment of radial Rydberg wave
packets produced by short laser pulses in the absence of external electric and
magnetic fields. Wave packets of this type are localized in the radial
coordinates and have p-state angular distributions. We argue that they can be
described by a particular analytical class of squeezed states, called radial
squeezed states. For hydrogenic Rydberg atoms, we discuss the time evolution of
the corresponding hydrogenic radial squeezed states. They are found to undergo
decoherence and collapse, followed by fractional and full revivals. We also
present their uncertainty product and uncertainty ratio as functions of time.
Our results show that hydrogenic radial squeezed states provide a suitable
analytical description of hydrogenic Rydberg atoms excited by short-pulsed
laser fields.Comment: published in Physical Review
Finding the center reliably: robust patterns of developmental gene expression
We investigate a mechanism for the robust identification of the center of a
developing biological system. We assume the existence of two morphogen
gradients, an activator emanating from the anterior, and a co-repressor from
the posterior. The co-repressor inhibits the action of the activator in
switching on target genes. We apply this system to Drosophila embryos, where we
predict the existence of a hitherto undetected posterior co-repressor. Using
mathematical modelling, we show that a symmetric activator-co-repressor model
can quantitatively explain the precise mid-embryo expression boundary of the
hunchback gene, and the scaling of this pattern with embryo size.Comment: 4 pages, 3 figure
Rydberg wavepackets in terms of hidden-variables: de Broglie-Bohm trajectories
The dynamics of highly excited radial Rydberg wavepackets is analyzed in
terms of de Broglie-Bohm (BB) trajectories. Although the wavepacket evolves
along classical motion, the computed BB trajectories are markedly different
from the classical dynamics: in particular none of the trajectories initially
near the atomic core reach the outer turning point where the wavepacket
localizes periodically. The reasons for this behavior, that we suggest to be
generic for trajectory-based hidden variable theories, are discussed.Comment: 12 pages, 4 fig
Forward Flux Sampling-type schemes for simulating rare events: Efficiency analysis
We analyse the efficiency of several simulation methods which we have
recently proposed for calculating rate constants for rare events in stochastic
dynamical systems, in or out of equilibrium. We derive analytical expressions
for the computational cost of using these methods, and for the statistical
error in the final estimate of the rate constant, for a given computational
cost. These expressions can be used to determine which method to use for a
given problem, to optimize the choice of parameters, and to evaluate the
significance of the results obtained. We apply the expressions to the
two-dimensional non-equilibrium rare event problem proposed by Maier and Stein.
For this problem, our analysis gives accurate quantitative predictions for the
computational efficiency of the three methods.Comment: 19 pages, 13 figure
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