scholarly journals Towards a full quantitative description of single-molecule reaction kinetics in biological cells

2018 ◽  
Vol 20 (24) ◽  
pp. 16393-16401 ◽  
Author(s):  
Denis S. Grebenkov ◽  
Ralf Metzler ◽  
Gleb Oshanin
Keyword(s):  
Single Molecule ◽  
First Passage Time ◽  
Quantitative Description ◽  
Threshold Value ◽  
Mathematical Concept ◽  
Passage Time ◽  
Molecule Reaction ◽  
Cylindrical Annulus ◽  
The Moment ◽  
First Time

The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. We present a robust explicit approach for obtaining the full distribution of FPT to a partially reactive target in a cylindrical-annulus domain.

scholarly journals First excess levels of vector processes

1994 ◽  
Vol 7 (3) ◽  
pp. 457-464 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Point Process ◽  
Renewal Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Compound Process ◽  
First Time

This paper analyzes the behavior of a point process marked by a two-dimensional renewal process with dependent components about some fixed (two-dimensional) level. The compound process evolves until one of its marks hits (i.e. reaches or exceeds) its associated level for the first time. The author targets a joint transformation of the first excess level, first passage time, and the index of the point process which labels the first passage time. The cases when both marks are either discrete or continuous or mixed are treated. For each of them, an explicit and compact formula is derived. Various applications to stochastic models are discussed.

scholarly journals The Protein Hourglass: Exact First Passage Time Distributions for Protein Thresholds

2020 ◽  
Author(s):  
Krishna Rijal ◽  
Ashok Prasad ◽  
Dibyendu Das
Keyword(s):  
First Passage Time ◽  
Self Regulation ◽  
Threshold Value ◽  
Passage Time ◽  
First Passage Times ◽  
Lambda Phage ◽  
First Passage ◽  
Analytical Expression ◽  
Protein Levels ◽  
Passage Times

Protein thresholds have been shown to act as an ancient timekeeping device, such as in the time to lysis of E. coli infected with bacteriophage lambda. The time taken for protein levels to reach a particular threshold for the first time is defined as the first passage time of the protein synthesis system, which is a stochastic quantity. The first few moments of the distribution of first passage times were known earlier, but an analytical expression for the full distribution was not available. In this work, we derive an analytical expression for the first passage times for a long-lived protein. This expression allows us to calculate the full distribution not only for cases of no self-regulation, but also for both positive and negative self-regulation of the threshold protein. We show that the shape of the distribution matches previous experimental data on lambda-phage lysis time distributions. We also provide analytical expressions for the FPT distribution with non-zero degradation in Laplace space. Furthermore, we study the noise in the precision of the first passage times described by coefficient of variation (CV) of the distribution as a function of the protein threshold value. We show that under conditions of positive self-regulation, the CV declines monotonically with increasing protein threshold, while under conditions of linear negative self-regulation, there is an optimal protein threshold that minimizes the noise in the first passage times.

Relating cellular signaling timescales to single-molecule kinetics: A first-passage time analysis of Ras activation by SOS

2021 ◽  
Vol 118 (45) ◽  
pp. e2103598118
Author(s):  
William Y. C. Huang ◽  
Steven Alvarez ◽  
Yasushi Kondo ◽  
John Kuriyan ◽  
Jay T. Groves
Keyword(s):  
Single Molecule ◽  
Cellular Signaling ◽  
First Passage Time ◽  
Passage Time ◽  
Nucleotide Exchange ◽  
Time Analysis ◽  
First Passage ◽  
Nucleotide Exchange Factor ◽  
Ras Activation ◽  
Son Of Sevenless

Son of Sevenless (SOS) is a Ras guanine nucleotide exchange factor (GEF) that plays a central role in numerous cellular signaling pathways. Like many other signaling molecules, SOS is autoinhibited in the cytosol and activates only after recruitment to the membrane. The mean activation time of individual SOS molecules has recently been measured to be ∼60 s, which is unexpectedly long and seemingly contradictory with cellular signaling timescales, which have been measured to be as fast as several seconds. Here, we rectify this discrepancy using a first-passage time analysis to reconstruct the effective signaling timescale of multiple SOS molecules from their single-molecule activation kinetics. Along with corresponding experimental measurements, this analysis reveals how the functional response time, comprised of many slowly activating molecules, can become substantially faster than the average molecular kinetics. This consequence stems from the enzymatic processivity of SOS in a highly out-of-equilibrium reaction cycle during receptor triggering. Ultimately, rare, early activation events dominate the macroscopic reaction dynamics.

A stochastic theoretical approach to study the size-dependent catalytic activity of a metal nanoparticle at the single molecule level

2017 ◽  
Vol 19 (13) ◽  
pp. 8889-8895 ◽  
Author(s):  
Divya Singh ◽  
Srabanti Chaudhury
Keyword(s):  
Catalytic Activity ◽  
Single Molecule ◽  
Time Distribution ◽  
First Passage Time ◽  
Theoretical Method ◽  
Passage Time ◽  
Metal Nanoparticle ◽  
First Passage ◽  
Single Molecule Level ◽  
Size Dependent

We present a theoretical method based on the first passage time distribution formalism to study the size-dependent catalytic activity of metal nanoparticle at the single molecule level.

scholarly journals MULTIFRACTAL MEASURES ON SMALL-WORLD NETWORKS

Fractals ◽  
2006 ◽  
Vol 14 (02) ◽  
pp. 119-123 ◽  
Author(s):  
K. H. CHANG ◽  
B. C. CHOI ◽  
SEONG-MIN YOON ◽  
KYUNGSIK KIM
Keyword(s):  
First Passage Time ◽  
Small World ◽  
Passage Time ◽  
Small World Network ◽  
Small World Networks ◽  
First Passage ◽  
One Dimensional ◽  
Random Walker ◽  
Regular Networks ◽  
First Time

We investigate the multifractals of the first passage time on a one-dimensional small-world network with reflecting and absorbing barriers. The multifractals can be obtained from the distribution of the first passage time at which the random walker arrives for the first time at an absorbing barrier after starting from an arbitrary initial site. Our simulation is found to estimate the fractal dimension D0 = 0.920 ~ 0.930 for the different network sizes and random rewiring fractions. In particular, the multifractal structure breaks down into a small-world network, when the rewiring fraction p is larger than the critical value pc = 0.3. Our simulation results are compared with the numerical computations for regular networks.

scholarly journals First passage time study of DNA strand displacement

2020 ◽  
Author(s):  
D. W. Bo Broadwater ◽  
Alexander W. Cook ◽  
Harold D. Kim
Keyword(s):  
Single Molecule ◽  
First Passage Time ◽  
Resonance Energy ◽  
Passage Time ◽  
Single Step ◽  
Strand Displacement ◽  
First Passage ◽  
Dna Strand Displacement ◽  
Dna Strand ◽  
Kinetics Of

AbstractDNA strand displacement, where a single-stranded nucleic acid invades a DNA duplex, is pervasive in genomic processes and DNA engineering applications. The kinetics of strand displacement have been studied in bulk; however, the kinetics of the underlying strand exchange were obfuscated by a slow bimolecular association step. Here, we use a novel single-molecule Fluorescence Resonance Energy Transfer (smFRET) approach termed the “fission” assay to obtain the full distribution of first passage times of unimolecular strand displacement. At a frame time of 4.4 ms, the first passage time distribution for a 14-nt displacement domain exhibited a nearly monotonic decay with little delay. Among the eight different sequences we tested, the mean displacement time was on average 35 ms and varied by up to a factor of 13. The measured displacement kinetics also varied between complementary invaders and between RNA and DNA invaders of the same base sequence except for T→U substitution. However, displacement times were largely insensitive to the monovalent salt concentration in the range of 0.25 M to 1 M. Using a one-dimensional random walk model, we infer that the single-step displacement time is in the range of ∼30 µs to ∼300 µs depending on the base identity. The framework presented here is broadly applicable to the kinetic analysis of multistep processes investigated at the single-molecule level.

scholarly journals First Passage Problems for Asymmetric Wiener Processes

2006 ◽  
Vol 43 (1) ◽  
pp. 175-184 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Wiener Process ◽  
Generating Function ◽  
First Passage Time ◽  
Moment Generating Function ◽  
Passage Time ◽  
Expected Value ◽  
First Passage ◽  
One Dimensional ◽  
Wiener Processes ◽  
The Moment

The problem of computing the moment generating function of the first passage time T to a > 0 or −b < 0 for a one-dimensional Wiener process {X(t), t ≥ 0} is generalized by assuming that the infinitesimal parameters of the process may depend on the sign of X(t). The probability that the process is absorbed at a is also computed explicitly, as is the expected value of T.

A Probabilistic Theory of Random Maps

2005 ◽  
Author(s):  
Ozer Elbeyli ◽  
J. Q. Sun
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Probabilistic Theory ◽  
Kolmogorov Equations ◽  
Random Maps ◽  
Continuous State ◽  
Time Problem ◽  
Fpk Equation ◽  
The Moment ◽  
First Passage Time Problem

We present a probabilistic theory of random maps with discrete time and continuous state. The forward and backward Kolmogorov equations as well as the FPK equation governing the evolution of the probability density function of the system are derived. The moment equations, the reliability and first passage time problem are studied. The present work compliments the existing theory of continuous time stochastic processes.

First-Passage-Time Prototypes for Precipitation Statistics

2014 ◽  
Vol 71 (9) ◽  
pp. 3269-3291 ◽  
Author(s):  
Samuel N. Stechmann ◽  
J. David Neelin
Keyword(s):  
Water Vapor ◽  
Power Law ◽  
First Passage Time ◽  
Threshold Value ◽  
Passage Time ◽  
Precipitation Event ◽  
Prototype Model ◽  
First Passage ◽  
Stratiform Rain ◽  
Time Problem

Abstract Prototype models are presented for time series statistics of precipitation and column water vapor. In these models, precipitation events begin when the water vapor reaches a threshold value and end when it reaches a slightly lower threshold value, as motivated by recent observational and modeling studies. Using a stochastic forcing to parameterize moisture sources and sinks, this dynamics of reaching a threshold is a first-passage-time problem that can be solved analytically. Exact statistics are presented for precipitation event sizes and durations, for which the model predicts a probability density function (pdf) with a power law with exponent −. The range of power-law scaling extends from a characteristic small-event size to a characteristic large-event size, both of which are given explicitly in terms of the precipitation rate and water vapor variability. Outside this range, exponential scaling of event-size probability is shown. Furthermore, other statistics can be computed analytically, including cloud fraction, the pdf of water vapor, and the conditional mean and variance of precipitation (conditioned on the water vapor value). These statistics are compared with observational data for the transition to strong convection; the stochastic prototype captures a set of properties originally analyzed by analogy to critical phenomena. In a second prototype model, precipitation is further partitioned into deep convective and stratiform episodes. Additional exact statistics are presented, including stratiform rain fraction and cloud fractions, that suggest that even very simple temporal transition rules (for stratiform rain continuing after convective rain) can capture aspects of the role of stratiform precipitation in observed precipitation statistics.

scholarly journals Dynamic Analysis of a Tumor-Immune System under Allee Effect

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Chunmei Zeng ◽  
Shaojuan Ma
Keyword(s):  
Immune System ◽  
Equilibrium Point ◽  
Allee Effect ◽  
Deterministic Model ◽  
First Passage Time ◽  
Stochastic Averaging ◽  
Threshold Value ◽  
Passage Time ◽  
Steady State Probability ◽  
Coexistence Equilibrium

In this paper, we develop a definite tumor-immune model considering Allee effect. The deterministic model is studied qualitatively by mathematical analysis method, including the positivity, boundness, and local stability of the solution. In addition, we explore the effect of random factors on the transition of the tumor-immune system from a stable coexistence equilibrium point to a stable tumor-free equilibrium point. Based on the method of stochastic averaging, we obtain the expressions of the steady-state probability density and the mean first-passage time. And we find that the Allee effect has the greatest impact on the number of cells in the system when the Allee threshold value is within a certain range; the intensity of random factors could affect the likelihood of the system crossing from the coexistence equilibrium to the tumor-free equilibrium.

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