Multiparticle kA→0 Reaction in Small Volumes
1. Introduction
The distinct feature of the cell dynamics, when compared to in vitro experiments, is that biochemical reactions are well organized in space [1]. The living cell is not a bag of enzymes where all molecules are lumped together. Intracellular reactions occur in finite volumes that can be rather small. For example, the diameter of a reaction volume ranges from few nm (Golgi apparatus, ribosomes) towards micrometers (cytoplasm). The basic transport mechanism in the living cell is diffusion. There are other mechanisms but the diffusion is the most frequent one. Thus understanding of the diffusion limited kinetics of few reactants in small volumes is of considerable interest.
In the reactions in restricted geometries context, there have been more studies of binary than non-binary reactions. Perhaps, the main reason is that non-binary reactions are rare in the cell cytoplasm. The likelihood of many particles meeting in the vicinity of each other at the same time is rather small. However, there are many instances where sequential reactions appear to be multiparticle like due to strong cooperativity effects [2]. From that point of view, the study of multiparticle reactions in small volumes should be relevant for understanding the living cell biochemistry.
The talk with discuss the basic features of diffusion controlled reactions in infinite and finite (small volumes), emergence of anomalous kinetics (breakdown of mean field equations). An example of the breakdown of the mean field equations will be shown [3,4] that render the mean field equation qualitatively wrong. Finally, the results of a recent study of a multiparticle reaction model [5] will be presented.
2. Multiparticle reaction model and Method
The model is defined as follows. Particles A react in clusters of size k with the reaction rate that depends on the positions of particles in a well defined way. Particle are spatially extended objects (not points) with radius a. The goal will be to investigate kinetics for varying degree of cooperativity k, and the size of the particles a. For a given reaction instance the reaction rate is not zero if all reacting particles in the cluster are close enough. It is sufficient that one of the particles is far away from the cluster and the reaction instance will not proceed to the completion.
The model has been solved analytically using the formalism of a field theory. The master equation has been mapped onto the corresponding quantum field theory using the standard techniques, and the equations of motion for many-particle density functions have been derived. To do the calculations the formalism described in Ref. [4] was used heavily. The (infinite) hierarchy of equations has been solved by truncating the hierarchy at the level of the (k + 1)-density function. The expression for the time dependent effective reaction rate has been found in an approximate form.
3. Conclusion
The equations of motion differ significantly from the ones that describe binary reactions. The average particle number decays exponentially for large times. All scales in the problem combine non-trivially into the effective decay rate, and it is hard to identify the scales that control onset of the exponential behaviour. An analytic expression for the decay rate will be given.
References
[1] L. Pagliaro, A Survey of Cell Biology 192 (2000) 303.
[2] J.E. Jr Ferrell, J. Biol. 8 (2009) doi:10.1186/jbiol157.
[3] Z. Konkoli, H. Johannesson, Phys. Rev. E 62 (2000) 3276.
[4] Z. Konkoli, Phys. Rev. E 69 (2004) 011106.
[5] Z. Konkoli, (2010) (Submitted).