139 research outputs found
Mean exit time for surface-mediated diffusion: spectral analysis and asymptotic behavior
We consider a model of surface-mediated diffusion with alternating phases of
pure bulk and surface diffusion. For this process, we compute the mean exit
time from a disk through a hole on the circle. We develop a spectral approach
to this escape problem in which the mean exit time is explicitly expressed
through the eigenvalues of the related self-adjoint operator. This
representation is particularly well suited to investigate the asymptotic
behavior of the mean exit time in the limit of large desorption rate .
For a point-like target, we show that the mean exit time diverges as
. For extended targets, we establish the asymptotic approach to
a finite limit. In both cases, the mean exit time is shown to asymptotically
increase as tends to infinity. We also revise the optimality regime
of surface-mediated diffusion. Although the presentation is limited to the unit
disk, the spectral approach can be extended to other domains such as rectangles
or spheres.Comment: 21 pages, 7 figure
Enhanced reaction kinetics in biological cells
The cell cytoskeleton is a striking example of "active" medium driven
out-of-equilibrium by ATP hydrolysis. Such activity has been shown recently to
have a spectacular impact on the mechanical and rheological properties of the
cellular medium, as well as on its transport properties : a generic tracer
particle freely diffuses as in a standard equilibrium medium, but also
intermittently binds with random interaction times to motor proteins, which
perform active ballistic excursions along cytoskeletal filaments. Here, we
propose for the first time an analytical model of transport limited reactions
in active media, and show quantitatively how active transport can enhance
reactivity for large enough tracers like vesicles. We derive analytically the
average interaction time with motor proteins which optimizes the reaction rate,
and reveal remarkable universal features of the optimal configuration. We
discuss why active transport may be beneficial in various biological examples:
cell cytoskeleton, membranes and lamellipodia, and tubular structures like
axons.Comment: 10 pages, 2 figure
On the Plants Leaves Boundary, "Jupe \`a Godets" and Conformal Embeddings
The stable profile of the boundary of a plant's leaf fluctuating in the
direction transversal to the leaf's surface is described in the framework of a
model called a "surface \`a godets". It is shown that the information on the
profile is encoded in the Jacobian of a conformal mapping (the coefficient of
deformation) corresponding to an isometric embedding of a uniform Cayley tree
into the 3D Euclidean space. The geometric characteristics of the leaf's
boundary (like the perimeter and the height) are calculated. In addition a
symbolic language allowing to investigate statistical properties of a "surface
\`a godets" with annealed random defects of curvature of density is
developed. It is found that at the surface exhibits a phase transition
with critical exponent from the exponentially growing to the flat
structure.Comment: 17 pages (revtex), 8 eps-figures, to appear in Journal of Physics
Random walk generated by random permutations of {1,2,3, ..., n+1}
We study properties of a non-Markovian random walk , , evolving in discrete time on a one-dimensional lattice of
integers, whose moves to the right or to the left are prescribed by the
\text{rise-and-descent} sequences characterizing random permutations of
. We determine exactly the probability of finding
the end-point of the trajectory of such a
permutation-generated random walk (PGRW) at site , and show that in the
limit it converges to a normal distribution with a smaller,
compared to the conventional P\'olya random walk, diffusion coefficient. We
formulate, as well, an auxiliary stochastic process whose distribution is
identic to the distribution of the intermediate points , ,
which enables us to obtain the probability measure of different excursions and
to define the asymptotic distribution of the number of "turns" of the PGRW
trajectories.Comment: text shortened, new results added, appearing in J. Phys.
Optimal search strategies for hidden targets
What is the fastest way of finding a randomly hidden target? This question of
general relevance is of vital importance for foraging animals. Experimental
observations reveal that the search behaviour of foragers is generally
intermittent: active search phases randomly alternate with phases of fast
ballistic motion. In this letter, we study the efficiency of this type of two
states search strategies, by calculating analytically the mean first passage
time at the target. We model the perception mecanism involved in the active
search phase by a diffusive process. In this framework, we show that the search
strategy is optimal when the average duration of "motion phases" varies like
the power either 3/5 or 2/3 of the average duration of "search phases",
depending on the regime. This scaling accounts for experimental data over a
wide range of species, which suggests that the kinetics of search trajectories
is a determining factor optimized by foragers and that the perception activity
is adequately described by a diffusion process.Comment: 4 pages, 5 figures. to appear in Phys. Rev. Let
Nuclear target search at the single molecule level: protein interactions define the exploration landscape
Gene regulation relies on highly mobile transcription factors (TFs) exploring the nucleoplasm in search of their targets. Our view of the nucleus has evolved from that of an isotropic and homogenous reactor to that of a highly organized yet very dynamic organelle. However important questions remain on how these regulatory factors explore the nuclear environment in search of their DNA or protein targets, and how their exploration strategy affects the kinetics of transcriptional regulation.
We implemented a single-molecule tracking assay to determine the TFs dynamics using photoactivatable tags in human cells. We investigated the mobility of several nuclear proteins, including the transcription factor c-Myc and the elongation factor P-TEFb. We found that, while their diffusion speed was comparable, these proteins largely differed in terms of their exploration geometry. We discovered that c-Myc is a global explorer diffusing in the nucleus without spatial constraints. In contrast, the positive transcription elongation factor P-TEFb is a local explorer that oversamples its environment, constrained by a fractal nuclear architecture. Consequently, each c-Myc molecule is equally available for all nuclear sites while P-TEFb reaches its targets in a position-dependent manner. We also measured the mobility of a P-TEFb mutant in which the interaction with the CTD of the RNA Pol II was truncated. In this case, the single-molecule experiments suggested a global exploration of the P-TEFb mutant, consistent with free diffusion.
Our observations are in line with a model in which the exploration geometry of TFs is constrained by their interactions and not by exclusion properties. Our findings have strong implications on how proteins react in the nucleus and how their function can be regulated in space and time
Classes of fast and specific search mechanisms for proteins on DNA
Problems of search and recognition appear over different scales in biological
systems. In this review we focus on the challenges posed by interactions
between proteins, in particular transcription factors, and DNA and possible
mechanisms which allow for a fast and selective target location. Initially we
argue that DNA-binding proteins can be classified, broadly, into three distinct
classes which we illustrate using experimental data. Each class calls for a
different search process and we discuss the possible application of different
search mechanisms proposed over the years to each class. The main thrust of
this review is a new mechanism which is based on barrier discrimination. We
introduce the model and analyze in detail its consequences. It is shown that
this mechanism applies to all classes of transcription factors and can lead to
a fast and specific search. Moreover, it is shown that the mechanism has
interesting transient features which allow for stability at the target despite
rapid binding and unbinding of the transcription factor from the target.Comment: 65 pages, 23 figure
Intermittent search strategies
This review examines intermittent target search strategies, which combine
phases of slow motion, allowing the searcher to detect the target, and phases
of fast motion during which targets cannot be detected. We first show that
intermittent search strategies are actually widely observed at various scales.
At the macroscopic scale, this is for example the case of animals looking for
food ; at the microscopic scale, intermittent transport patterns are involved
in reaction pathway of DNA binding proteins as well as in intracellular
transport. Second, we introduce generic stochastic models, which show that
intermittent strategies are efficient strategies, which enable to minimize the
search time. This suggests that the intrinsic efficiency of intermittent search
strategies could justify their frequent observation in nature. Last, beyond
these modeling aspects, we propose that intermittent strategies could be used
also in a broader context to design and accelerate search processes.Comment: 72 pages, review articl
Banding, Excitability and Chaos in Active Nematic Suspensions
Motivated by the observation of highly unstable flowing states in suspensions
of microtubules and kinesin, we analyze a model of mutually-propelled filaments
suspended in a solvent. The system undergoes a mean-field isotropic-nematic
transition for large enough filament concentrations when the nematic order
parameter is allowed to vary in space and time. We analyze the model in two
contexts: a quasi-one-dimensional channel with no-slip walls and a
two-dimensional box with periodic boundaries. Using stability analysis and
numerical calculations we show that the interplay between non-uniform nematic
order, activity, and flow results in a variety of complex scenarios that
include spontaneous banded laminar flow, relaxation oscillations, and chaos.Comment: 15 pages, 15 figure
Geometry-controlled kinetics
It has long been appreciated that transport properties can control reaction
kinetics. This effect can be characterized by the time it takes a diffusing
molecule to reach a target -- the first-passage time (FPT). Although essential
to quantify the kinetics of reactions on all time scales, determining the FPT
distribution was deemed so far intractable. Here, we calculate analytically
this FPT distribution and show that transport processes as various as regular
diffusion, anomalous diffusion, diffusion in disordered media and in fractals
fall into the same universality classes. Beyond this theoretical aspect, this
result changes the views on standard reaction kinetics. More precisely, we
argue that geometry can become a key parameter so far ignored in this context,
and introduce the concept of "geometry-controlled kinetics". These findings
could help understand the crucial role of spatial organization of genes in
transcription kinetics, and more generally the impact of geometry on
diffusion-limited reactions.Comment: Submitted versio
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