TY  - INPR
A1  - Keller, Peter
T1  - Mathematical modeling of molecular motors
N2  - Amongst the many complex processes taking place in living cells, transport of cargoes across the cytosceleton is fundamental to cell viability and activity. To move cargoes between the different cell parts, cells employ Molecular Motors. The motors operate by transporting cargoes along the so-called cellular micro-tubules, namely rope-like structures that connect, for instance, the cell-nucleus and outer membrane. We introduce a new Markov Chain, the killed Quasi-Random-Walk, for such transport molecules and derive properties like the maximal run length and time. Furthermore we introduce permuted balance, which is a more flexible extension of the ordinary reversibility and introduce the notion of Time Duality, which compares certain passage times pathwise. We give a number of sufficient conditions for Time Duality based on the geometry of the transition graph. Both notions are closely related to properties of the killed Quasi-Random-Walk.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 1 
KW  - Markov chain
KW  - time duality
KW  - transition path theory
KW  - absorption
KW  - molecular motor
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63045
ER  - 
TY  - INPR
A1  - Keller, Peter
A1  - Roelly, Sylvie
A1  - Valleriani, Angelo
T1  - A quasi-random-walk to model a biological transport process
N2  - Transport Molecules play a crucial role for cell viability. Amongst others, linear motors transport cargos along rope-like structures from one location of the cell to another in a stochastic fashion. Thereby each step of the motor, either forwards or backwards, bridges a fixed distance. While moving along the rope the motor can also detach and is lost. We give here a mathematical formalization of such dynamics as a random process which is an extension of Random Walks, to which we add an absorbing state to model the detachment of the motor from the rope. We derive particular properties of such processes that have not been available before. Our results include description of the maximal distance reached from the starting point and the position from which detachment takes place. Finally, we apply our theoretical results to a concrete established model of the transport molecule Kinesin V.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 3 
KW  - Markov chain
KW  - random walk
KW  - molecular motor
KW  - step process
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63582
ER  - 
TY  - JOUR
A1  - Keller, Peter
A1  - Roelly, Sylvie
A1  - Valleriani, Angelo
T1  - On time duality for Markov Chains
JF  - Stochastic models
N2  - For an irreducible continuous time Markov chain, we derive the distribution of the first passage time from a given state i to another given state j and the reversed passage time from j to i, each under the condition of no return to the starting point. When these two distributions are identical, we say that i and j are in time duality. We introduce a new condition called permuted balance that generalizes the concept of reversibility and provides sufficient criteria, based on the structure of the transition graph of the Markov chain. Illustrative examples are provided.
KW  - Time duality
KW  - Detailed balance
KW  - First passage time
KW  - Reversibility
KW  - Permuted balance
KW  - Markov chain
Y1  - 2015
U6  - https://doi.org/10.1080/15326349.2014.969736
SN  - 1532-6349
SN  - 1532-4214
VL  - 31
IS  - 1
SP  - 98
EP  - 118
PB  - Taylor & Francis Group
CY  - Philadelphia
ER  -