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In AC, we primarily consider the qualitative aspects of a problem, before considering the quantitative relations between its components. The quantitative aspect is usually analyzed using reactor flow equations Yock92 . The stable structures generated by artificial chemistries, the stable sets of molecules, are usually ... |
Some of the aspects very commonly studied in AC are - given an AC, how to know a priori, which organizations are possible and which are not possible? To know which organizations are probable and which are improbable? To define an AC to generate a particular organization? How stable are organizations? Can the complexity... |
Ditt01 has detailed description of several interesting common phenomena which are observed in different kinds of AC systems such as reduction of diversity, formation of densely coupled stabled networks, syntactic and semantic closure in these networks etc. |
P System based Artificial Graph Chemistry In the previous section we reviewed some examples of ACs with discussion on their positive and negative aspects from the point of view of pre-biotic evolution of life ranging from the level of abstractions of essential molecular properties from real chemistry to the nature of e... |
In this section we will propose a new AC to address the problem of lack of structural abstraction in molecular structures and reaction rules. Again as discussed in the previous section topological constraints play significant role in emergence of certain phenomena in ACs, for example emergence of membrane type structur... |
2.1 P System G. Paun Paun00 (ref Paun02 ) introduced membrane system as a model of parallel and distributed computation with “membrane” type structure. P system is a basic model of a membrane systems with membranes arranged in a hierarchial structure, as in a cell, and processing multisets of symbol-objects. |
Definition The main components of a membrane system are the membrane structure multisets of objects , and the evolution rules . A membrane structure describes the mutual relationship between membranes, the relations of adjacency, of being in and out. |
Formally a membrane structure is defined as follows. Consider a context free language [MATH] defined over the alphabet [MATH] and |
[EQUATION] Then language [MATH] over alphabet {} is defined as [EQUATION] that is, MS consists of any string of correctly matching pair of parentheses with a matching pair at the end. [MATH] is one such example where brackets are numbered (with subscripts) to match the correct pairing. |
Let [MATH] be the set of all equivalence classes of [MATH] with respect to reflexive and transitive closure of the relation [MATH] defined over [MATH] such that for [MATH] [MATH] |
if and only if [MATH] and [MATH] can be made same when two pairs of parentheses which are not contained in each other are interchanged, together with their contents. The elements of |
[MATH] are called membrane structures Each matching pair of parentheses is called a membrane There is a unique external membrane called skin . For a membrane structure [MATH] , any closed space delimited by a membrane is called a region of [MATH] . Alternately if we consider a membrane structure as a directed unordered... |
regions, one associated with each membrane. Consider [MATH] as denumerable set of objects. Let [MATH] be a membrane structure of degree [MATH] , with the membranes labelled in a one to one manner, for instance, with the numbers from [MATH] to [MATH] . In this way, the regions of [MATH] are also identified by the number... |
In other words a super-cell is defined as [MATH] , where V is an alphabet; its elements are called objects. Super cell extended to a P system of degree [MATH] [MATH] , is a construct |
[EQUATION] where: (i) [MATH] is a membrane structure of degree [MATH] , with the membranes and the regions labelled in a one to one manner with elements in a given set [MATH] , which can be assumed to be just |
[MATH] (ii) [MATH] [MATH] , are multisets over [MATH] associated with the regions [MATH] of [MATH] (iv) [MATH] [MATH] , are finite sets of evolution rules over [MATH] . An evolution rule is of the form [MATH] where [MATH] is a string over [MATH] and [MATH] , where |
[MATH] is a string over [EQUATION] and [MATH] is a special symbol not in [MATH] (v) [MATH] is a number between [MATH] and [MATH] which specifies the output membrane of [MATH] |
The symbols [MATH] [MATH] [MATH] [MATH] are called target commands or target indicators . Length of [MATH] in [MATH] is called the radius of the rule. These rules are explained in the next section. |
Evolution of a P System The above defined P system evolves as follows. Consider a rule [MATH] in a set [MATH] . We look to the region associated with the membrane [MATH] . If the objects mentioned by [MATH] , with the multiplicities specified by [MATH] , appear in [MATH] , then these objects can evolve according to the... |
The result of using the rule is determined by [MATH] . If an object appears in [MATH] in a pair [MATH] , then it will remain in the same region [MATH] . If an object appears in [MATH] in a pair [MATH] , then [MATH] will exit the membrane [MATH] and will become an element of the region immediately outside it (thus, it w... |
[MATH] is not one of the membranes delimiting “from below” the region [MATH] , then the application of the rule is not allowed. If the symbol [MATH] appears in [MATH] , then the membrane [MATH] is removed (said dissolved) and at the same time the set of rules |
[MATH] is removed. The multiset [MATH] is added (in the sense of multisets union) to the multiset associated with the region which was immediately external to the membrane [MATH] . We do not allow the dissolving of the skin, the outermost membrane because this means that the super cell is lost, we do no longer have a c... |
A P system evolves as a whole unit with all possible applications of rules at the same time. That means, all the operations are done in parallel, for all possible applicable rules [MATH] for all occurrences of multisets specified by [MATH] in the region associated with the rule and all regions are considered at the sam... |
If there are rules in a super cell system [MATH] with the radius at least two, then the system is said to be cooperative ; in the opposite case, it is called non cooperative . A system is said to be catalytic if there are certain objects [MATH] specified in advance, called catalysts , such that the rules of the system ... |
Biological analogy of Super Cell system. The mode of evolving of objects in a super cell provided with evolution rules as described above can be interpreted in the following - idealized - biochemical way. We have an organism, delimited by a skin (the skin membrane). Inside, there are free molecules, organized hierarchi... |
2.2 Design of the Chemistry Having defined the basic concepts of P System, we will now discuss the basic components of the P system based artificial graph chemistry (AGC) by defning an extension of the P System and the molucules as labelled graph. |
2.2.1 A Probabilistic Extension of P System We extend a P system by associating probabilities with rules. These probabilities can be interpreted as relative frequencies of rule applications on the molecules over the course of evolution. This happens in real chemistries as well, where in a large pool of several chemical... |
Another main modification on the basic definition of P system is that to suit an AC set up, we modify the basic P system with multisets of symbols and rewriting rules to a P system with multisets of molecules with the reaction rules. The structure of molecules with reaction semantics is presented next. |
2.2.2 Molecular Structure Representation and Reaction Semantics We represent a molecule as undirected labelled graph and develop reaction semantics to represent molecular reactions. The main motivation behind the selection of a graph comes from the observation on the lack of the structural abstractions in ACs like lamb... |
Formally a molecule is represented as a undirected labelled graph [MATH] with mapping [MATH] associating weights with edges [MATH] , where [MATH] is the set of real numbers. [MATH] can also be represented as weighted symmetric matrix. Each node of the graph can be thought of as an “atom” and an edge as “chemical bond” ... |
Each reaction rule is a mapping from a subset of molecules called reactants to another subset of molecules called products . In other words a reaction can be thought of as a [MATH] -ary ( [MATH] graph transformation operator . Reactions are constrained by |
guards , which are used to capture the thermodynamic conditions controlling the real chemical reactions. We can consider two cases. In the first case the set of possible reaction rules is fixed and determined priori. Only the guards are evaluated later. Secondly we can have some generic laws controlling the nature of p... |
Initially we choose no specific spatial structure on distribution of molecules. Thus in essence each molecule can potentially react with any other molecule. The dynamics is controlled stochastically, that is, we randomly select the rules and see if they can be carried out, if yes, we select random concentration of the ... |
A spatial structure can be imposed on the molecules using our extended probabilistic P-system, where sets of molecules will be encapsulated inside membranes for local reactions, with possible migration of molecules across membranes. |
Final Discussion Objective of this section is to summarize main goals of the proposal and discuss the broader picture where these goals may fit in an AC research. |
To summarize, we defined a probabilistic P system based AGC with the aim of understanding the principles leading to the evolution of life-like structures in an AC set up. We need to explore it further by carrying out detailed experiments with varying parameters such as definitions of reaction rules, presence and nature... |
ACs are basically designed to complement the main stream AL research. This is primarily because major AL studies presume the prior existence of basic structure of life-like entities and develop over them. This leaves the question of origin of these basic structures open and that is where ACs come into picture. |
Because the main theme of AL research is to discover the possible biological principles which might be working independent of physical laws, AL studies mainly draw motivation from real-life biological phenomena. Theory of evolution based on random mutations and fitness based natural selection is one such source of moti... |
This is where this proposal is expected to contribute most by explaining issues like what are the fundamental ingredients of an AC set up which will lead to interesting emergent organizations? Can these ingredients be related to information and/or computation? Is an AC able to create information? Does “information proc... |
# Source: arxiv 0901.0350 # Title: Properties of tug-of-war model for cargo transport by molecular motors # Sections: all # Downloaded: 2026-03-03T05:14:41.817967+00:00 |
Properties of tug-of-war model for cargo transport by molecular motors Abstract Molecular motors are essential components for the biophysical functions of the cell. Current quantitative understanding of how multiple motors move along a single track is not complete, even though models and theories for a single motor mec... |
have developed a tug-of-war model to describe the bidirectional movement of the cargo (PNAS(2008) 105(12) P4609-4614). They found that the tug-of-war model exhibits several qualitative different motility regimes, which depend on the precise value of single motor parameters, and they suggested the sensitivity can be use... |
PACS : 87.16.Nn, 87.16.A-, 82.39.-k, 05.40.Jc Keywords : Tug-of-war, molecular motors, intracellular transport Introduction Molecular motors, including biological motor proteins such as kinesin |
, dynein , mysion and [MATH] -ATP synthase , are mechanochemical force generators which convert chemical or biochemical energy in the form of chemical potential into mechanical work in thermal environment |
. The mechanochemical process is accomplished by individual macromolecules, immersed in an aqueous solution with the chemical potential, moving along a linear track. Many biological motor proteins move processively. For example, myosin slides along an actin filament, kinesin and dynein along microtubule (MT). All of th... |
): Kinesin moves towards the plus end of the MT and dynein towards the minus end. In comparison with the macroscopic engines driven by Carnot cycles, molecular motors have a high energy efficiency at about 50%, while the energy efficiency of a car is about 15%-20% |
. Furthermore, the velocities of molecular motors are also fast with mean velocity be at about several hundreds nanometers per second |
. However, the most significant difference between the molecular motors and the macroscopic engines is that the former are moving in a thermal noise dominated environment |
. So the movement of the molecular motors should be described stochastically, rather than determinately. Being able to convert and harvest energy with high efficiency on a mesoscopic scale makes molecular motors an exciting area of scientific research with potentially great innovative applications for energy production... |
Great progress has been made in recent years in modeling the movement of molecular motors, including the mean field methods , the Langevin stochastic dynamic methods |
and discrete stochastic methods . However, the existing models for a single molecular motor are not sufficient in predicting the recent experimental results: It is found that bidirectional motion of the cargo, which is carried by motor proteins, exhibits different patterns in different stages of embryonic development( |
). Following these recent experimental results ( ), Lipowsky and his coworkers have developed the tug-of-war model for describing the movement of the cargo carried by processive motors, such as kinesin and dynein |
). In their model, the experimentally known single motor properties are taken into account, so it is consistent with almost all experimental observations and can make quantitative predictions for bidirectional transport of the cargo. Since cargo movement carried by a single motor protein via an elastic tether has been ... |
, the focus of tug-of-war model is not on the detailed movement of cargo carried by a single motor per se , rather it concerns with the competition and cooperation of multiple motors on a single track (see the schematic depiction in Fig. ). |
In the present paper, we will give a further comprehensive mathematical analysis of tug-of-war model. Through detailed analysis, we find that the steady state movement of cargo is determined by the initial numbers of the two motor species which bound to the track of movement. Biophysically, the steady state is the only... |
The tug-of-war model The tug-of-war model is developed by Reinhard Lipowsky’s study group to study the bidirectional transport of the cargo, in which the cargo is attached with [MATH] plus and [MATH] minus motors. Particularly, if [MATH] or [MATH] , it recovers the usual model for cooperate transport by a single motor ... |
). In this model, each motor species is characterized by six parameters, which can be measured in single molecular experiments (see Tab. ): (i) stall force [MATH] |
(pN) (ii) detachment force [MATH] (pN) (iii) unbinding rate [MATH] [MATH] ) (iv) binding rate [MATH] [MATH] ) (v) forward velocity [MATH] [MATH] m/s) and (vi) superstall velocity amplitude [MATH] (nm/s). The motors bind to or unbind from a MT in a stochastic fashion, so that the cargo is pulled by [MATH] plus and [MATH... |
In tug-of-war model, it is assumed that, at every time [MATH] , the state of cargo with [MATH] plus and [MATH] minus motors firmly attached to it is fully characterized by numbers [MATH] and [MATH] of plus and minus motors that are bound to the MT. The state of cargo changes when a plus or a minus motor binds or unbind... |
[EQUATION] where [MATH] is the binding rate of a single plus (minus) motor to the MT, which depends only weakly on the load ) and therefore is taken equal to zero-load binding rate [MATH] [MATH] is the unbinding rate of a single plus (minus) motor from the MT, which increases exponentially with the applied force [MATH] |
[EQUATION] as measured for kinesin , where [MATH] is the detachment force. The governing equations for [MATH] or [MATH] are similar as ( ) except |
[MATH] and [MATH] Under the assumptions that the motors act independently and feel each other only due to two effects: (i) opposing motors act as load, and (ii) identical motors share this load, Lipowsky and coworkers gave the following relation (see |
[EQUATION] where [MATH] is the load felt by each plus (minus) motor. Eqs. ) ( ) imply [EQUATION] Here, the cargo force [MATH] is determined by the condition that plus motors, which experience the force [MATH] , and minus motors, which experience the force [MATH] , move with the same velocity |
[MATH] , which is the cargo velocity: [EQUATION] The same as in , the following piecewise linear force-velocity relation of a single motor is used in this paper: |
[EQUATION] where [MATH] is the absolute value of the superstall velocity amplitude, [MATH] is the zero-load forward velocity, [MATH] is the stall force. |
The velocity of cargo and unbinding rates of motors For the convenience of analysis in the following sections, we give the formulations of velocity of cargo and unbinding rates of plus and minus motors in this section. |
(I) In case of “stronger plus motors”, i.e. [MATH] , Eqs. ( ) ( ) lead to the cargo force and velocity: [EQUATION] Using Eqs. ( ) ( ), the unbinding rates of plus and minus motors are: |
[EQUATION] where [EQUATION] Let [MATH] [MATH] and [MATH] , then [EQUATION] (II) In case of “stronger minus motors”, i.e. [MATH] , the cargo force and velocity are: |
[EQUATION] Similar as in (I) , the unbinding rates of plus and minus motors are [EQUATION] in which [EQUATION] The splitting boundary of case (I) and case (II) is |
[MATH] , i.e. [MATH] (III) If an external force [MATH] is present, here [MATH] is taken to be positive if it points into the minus direction, then the force balance ( ) becomes |
[EQUATION] In case of [MATH] , carrying through the same calculation as for the case without external force leads to the velocity of cargo |
[EQUATION] The corresponding unbinding rates of the plus and minus motors are [EQUATION] (IV) If an external force [MATH] is present and |
[MATH] , then the velocity of cargo is [EQUATION] and the unbinding rates of plus and minus motors are [EQUATION] Similarly, the splitting boundary of case (III) and case (IV) is [MATH] , i.e. |
[MATH] (V) More generally, if there exists an external force [MATH] and the friction coefficient of cargo is [MATH] then in the case of [MATH] , the velocity of the cargo is |
[EQUATION] and the unbinding rates of plus and minus motors are [EQUATION] On the other hand, if [MATH] , then the velocity of cargo is |
[EQUATION] and the unbinding rates of plus and minus motors are [EQUATION] The splitting boundary of these two cases is also [MATH] , i.e. |
[MATH] The dynamics of motor numbers [MATH] and [MATH] For the sake of convenience, let [EQUATION] and [MATH] . During time interval [MATH] , the increase of plus motor number is |
[EQUATION] In the limit [MATH] , ( 23 ) leads to [EQUATION] Similarly, the dynamics of minus motor number is [EQUATION] So [MATH] satisfy |
[EQUATION] As we all know, the steady state solutions [MATH] of the system ( 26 ), which satisfy [MATH] and [MATH] , are stable if and only if the real parts of the two eigenvalues of the following matrix |
[EQUATION] are nonpositive. It is to say that [EQUATION] To better understanding, the figures of functions [MATH] are plotted in figure |
In view of conditions ( 28 ), to initial values [MATH] [MATH] , if the point [MATH] lies in the subdomain I (II or III), then the final state is stable steady state [MATH] [MATH] or [MATH] ) (see Fig. ). Theoretically, [MATH] [MATH] , but they are small than the accuracy of the numerical calculation used in this paper,... |
To further understand the properties of the stable steady state points, the figures of [MATH] and [MATH] with different values of parameters [MATH] [MATH] [MATH] |
[MATH] [MATH] [MATH] [MATH] [MATH] and [MATH] are plotted in Fig. and From the figures, one can find that system ( 26 ) might have one, two or three stable steady states, which depends on the values of the parameters. Given the initial value [MATH] , the final steady state can be determined using the similar method as ... |
Right ). One can be easily know that, almost all of the parameters used in the tug-of-war model have one or two critical points, the final stable steady state would change if one of the parameters jumps from one side of its critical points to another side. |
Obviously, for [MATH] or [MATH] (i.e. [MATH] or [MATH] ), the tug-of-war model is reduced to the usual model for cooperate transport by a single motor species (minus or plus), and the only stable steady state is [MATH] for plus motor species or [MATH] for minus motor species. The average velocity of the cargo at steady... |
[MATH] if [MATH] , and [MATH] if [MATH] , which are the velocities of a single motor. Comparison with Monte Carlo simulations Due to the above discussion, in large motor numbers limit [MATH] , the movement of the cargo might have one, two or three stable steady states. The final steady state is determined by the initia... |
For example, in case of Fig. (right), if [MATH] lies in subdomains (II) , the final steady state would be [MATH] However, if the numbers [MATH] of molecular motors, which attached to the cargo, is finite or even small, the steady states numbers [MATH] and [MATH] might be different with the theoretical values [MATH] and... |
[MATH] would lie in the neighborhoods of the theoretical values [MATH] and [MATH] . But, in small [MATH] cases, the steady state motor numbers [MATH] and [MATH] can jump easily from the neighborhood of one of the theoretical stable steady state point |
[MATH] to the neighborhood of another theoretical stable steady state point [MATH] (see Fig. ). For finite motor numbers [MATH] , the stepsize of the system 26 ) are [MATH] [MATH] . So the smaller of motor numbers [MATH] , the easier for motor numbers |
[MATH] to jump from one of the steady subdomains I, II or III to another. Intuitively, the probability that [MATH] lies in the neighborhood of the stable steady state point [MATH] is proportional to the area of [MATH] ’s steady subdomain. Mathematically, the probability of motor numbers [MATH] change from [MATH] to [MA... |
along trajectory [MATH] is [EQUATION] where [MATH] [MATH] if and only if [MATH] [MATH] if and only if [MATH] [MATH] if and only if [MATH] [MATH] if and only if |
[MATH] . So, theoretically, we can obtain the probability that motor numbers [MATH] change from the neighborhood of one stable steady states to the neighborhood of another stable steady states. From these transition probabilities, we can know more details about the steady state movement of the cargo in this small [MATH... |
Conclusion and remarks In this paper, the steady state properties of the recent tug-of-war model, which is provided by Lipowsky et al to model the movement of cargo, which is transported by two motor species in the cell, is discussed. Biophysically, the stable steady states are the most important states, because the tr... |
and ), so almost all of the data are measured in stable steady states. Through the discussion in this paper, we can know that the final steady states of the movement of the cargo is determined by initial numbers of the plus and minus motors which are bounded to the microtubule. Certainly, the velocity and direction of ... |
[MATH] and [MATH] , can be determined by the biochemical environment and properties of the cargos, so some of which can be transported from the plus end to the minus end, and others can be transported reversely. |
Acknowledgments This work was funded by National Natural Science Foundation of China (Grant No. 10701029). The author thanks professor Hong Qian of University of Washington for his help to complete this research. The author also thanks the reviewers for their help to improve the quality of this paper. |
# Source: arxiv 0901.0936 # Title: Dynamics of the bacterial flagellar motor with multiple stators # Sections: all # Downloaded: 2026-03-03T05:14:35.058582+00:00 |
Dynamics of the bacterial flagellar motor with multiple stators Giovanni Meacci and Yuhai Tu IBM T. J. Watson Research Center P.O. Box 218, Yorktown Heights, NY 10598 |
Corresponding author (email: yuhai@us.ibm.com) Abstract The bacterial flagellar motor drives the rotation of flagellar filaments and enables many species of bacteria to swim. Torque is PNAS 105, 1182-1185] show that near zero load the speed of the motor is independent of the number of stators. Here, we introduce a math... |
The swimming motion of bacterium Escherichia coli is propelled by the concerted rotational motion of its flagellar filaments BA73 B03 . Each filament ( [MATH] m long) is driven by a rotatory motor embedded in the cell wall, with a angular speed of the order of [MATH] Hz B03 . The motor has one rotor and multiple stator... |
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