Motivated by the application of a stochastic particle-based simulation model to study finite coagulation-fragmentation systems, an approximate coagulation-fragmentation model is introduced for the dynamics. Traditional large and small population examples are explored theoretically, as well as numerically, and approximate models are compared to traditional well-known coagulation-fragmentation frameworks. Our new approximate coagulation-fragmentation models are compared to traditional descriptions both in large population dynamical systems settings, as well as in small population Mean-First assembly time (MFAT) contexts. Stochastic simulations in the form of Gillespie's Stochastic Simulation Algorithm (SSA) for exact and approximate coagulation-fragmentation dynamics, as well as Reactive Multiparticle Collision Dynamics (RMPC) for particle-based versions thereof, are provided, as well as a numerical matrix inversion method for larger systems that is helpful to obtain theoretical MFAT estimates for larger populations. We also consider differences between a chemical system interpretation of the dynamics, Becker-Döring models, and Smoluchowski dynamics. As conventionally explored, formation and break-up rates in the form of constant, additive and multiplicative rates are considered, as well as large initial monomer-only conditions. Finally, a new time-rescaling mechanism is introduced so that MFAT results for RMPC simulations (for both exact and approximate models) can be obtained when the MFAT values are comparable to, or less than, an RMPC time step. For the traditional break-up rates considered in this paper, our findings show that MFATs are fairly comparable for the exact and approximate systems, and that an approximate RMPC model can be used in this context. Since RMPC is a stochastic simulation algorithm that explicitly incorporates positions and velocities of particles in the system, velocity-dependent reaction rates can be easily added to the reaction dynamics, as well as diffusive effects. Since RMPC can be used as a particle-based fluid-flow solver, and since our new approximate self-assembly process can lead to numerical speed-up of the self-assembly process in large population flow-problems, our results thus provide the stepping stone for a simulation paradigm that is capable of furthering our understanding of how self-assembly processes may be affected by diffusion or when subjected to flow conditions. The extent of the performance enhancement and accuracy of the approximate RMPC method applied to the self-assembly process in flow conditions has yet to be explored and is part of future work.
Citation: Pradeep Kunwar, Katrin Rohlf. Mean first assembly times in coagulation-fragmentation systems: Theoretical, stochastic, and approximate approaches[J]. AIMS Mathematics, 2026, 11(9): 31476-31515. doi: 10.3934/math.20261242
Motivated by the application of a stochastic particle-based simulation model to study finite coagulation-fragmentation systems, an approximate coagulation-fragmentation model is introduced for the dynamics. Traditional large and small population examples are explored theoretically, as well as numerically, and approximate models are compared to traditional well-known coagulation-fragmentation frameworks. Our new approximate coagulation-fragmentation models are compared to traditional descriptions both in large population dynamical systems settings, as well as in small population Mean-First assembly time (MFAT) contexts. Stochastic simulations in the form of Gillespie's Stochastic Simulation Algorithm (SSA) for exact and approximate coagulation-fragmentation dynamics, as well as Reactive Multiparticle Collision Dynamics (RMPC) for particle-based versions thereof, are provided, as well as a numerical matrix inversion method for larger systems that is helpful to obtain theoretical MFAT estimates for larger populations. We also consider differences between a chemical system interpretation of the dynamics, Becker-Döring models, and Smoluchowski dynamics. As conventionally explored, formation and break-up rates in the form of constant, additive and multiplicative rates are considered, as well as large initial monomer-only conditions. Finally, a new time-rescaling mechanism is introduced so that MFAT results for RMPC simulations (for both exact and approximate models) can be obtained when the MFAT values are comparable to, or less than, an RMPC time step. For the traditional break-up rates considered in this paper, our findings show that MFATs are fairly comparable for the exact and approximate systems, and that an approximate RMPC model can be used in this context. Since RMPC is a stochastic simulation algorithm that explicitly incorporates positions and velocities of particles in the system, velocity-dependent reaction rates can be easily added to the reaction dynamics, as well as diffusive effects. Since RMPC can be used as a particle-based fluid-flow solver, and since our new approximate self-assembly process can lead to numerical speed-up of the self-assembly process in large population flow-problems, our results thus provide the stepping stone for a simulation paradigm that is capable of furthering our understanding of how self-assembly processes may be affected by diffusion or when subjected to flow conditions. The extent of the performance enhancement and accuracy of the approximate RMPC method applied to the self-assembly process in flow conditions has yet to be explored and is part of future work.
| [1] | N. Hoze, D. Holcman, Kinetics of aggregation with a finite number of particles and application to viral capsid assembly, J. Math. Biol., 70 (2015), 1685–1705. https://doi.org/10.1007/s00285-014-0819-2 |
| [2] | S. Chandrasekhar, Stochastic problems in physics and astrophysics, Rev. Mod. Phys., 15 (1943), 1–89. https://doi.org/10.1103/RevModPhys.15.1 |
| [3] | N. Hoze, D. Holcman, Coagulation-fragmentation for a finite number of particles and application to telomere clustering in the yeast nucleus, Phys. Lett. A, 376 (2012), 845–849. https://doi.org/10.1016/j.physleta.2012.01.014 |
| [4] |
R. Yvinec, M. R. D'Orsogna, T. Chou, First passage times in homogeneous nucleation and self-assembly, J. Chem. Phys., 137 (2012), 244107. https://doi.org/10.1063/1.4772598 doi: 10.1063/1.4772598
|
| [5] |
M. R. D'Orsogna, G. Lakatos, T. Chou, Stochastic self-assembly of incommensurate clusters, J. Chem. Phys., 136 (2012), 084110. https://doi.org/10.1063/1.3688231 doi: 10.1063/1.3688231
|
| [6] |
R. Becker, W. Döring, Kinetische Behandlung der Keimbildung in übersättigten Dämpfern, Ann. Phys. (Leipzig), 24 (1935), 719–752. https://doi.org/10.1002/andp.19354160806 doi: 10.1002/andp.19354160806
|
| [7] | M. von Smoluchowski, Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen, Z. Phys., 17 (1916), 557–585. |
| [8] | P. L. Krapivski, S. Redner, E. Ben-Naim, A Kinetic view of statistical physics, Cambridge: Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511780516 |
| [9] |
D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists, Bernoulli, 5 (1999), 3–48. https://doi.org/10.2307/3318611 doi: 10.2307/3318611
|
| [10] |
M. R. D'Orsogna, Q. Lei, T. Chou, First assembly times and equilibration in stochastic coagulation-fragmentation, J. Chem. Phys., 143 (2015), 014112. https://doi.org/10.1063/1.4923002 doi: 10.1063/1.4923002
|
| [11] |
D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340–2361. https://doi.org/10.1021/j100540a008 doi: 10.1021/j100540a008
|
| [12] |
K. Tucci, R. Kapral, Mesoscopic model for diffusion-influenced reaction dynamics, J. Chem. Phys., 120 (2004), 8262–8270. https://doi.org/10.1063/1.1690244 doi: 10.1063/1.1690244
|
| [13] |
K. Tucci, R. Kapral, Mesoscopic multiparticle collision dynamics of reaction-diffusion fronts, J. Phys. Chem. B, 109 (2005), 21300–21304. https://doi.org/10.1021/jp052701u doi: 10.1021/jp052701u
|
| [14] |
K. Rohlf, S. Fraser, R. Kapral, Reactive multiparticle collision dynamics, Comput. Phys. Commun., 179 (2008), 132–139. https://doi.org/10.1016/j.cpc.2008.01.027 doi: 10.1016/j.cpc.2008.01.027
|
| [15] |
K. Rohlf, Stochastic phase-space description for reactions that change particle numbers, J. Math. Chem., 45 (2009), 141–160. https://doi.org/10.1007/s10910-008-9373-8 doi: 10.1007/s10910-008-9373-8
|
| [16] |
R. Strehl, K. Rohlf, Multiparticle collision dynamics for diffusion-influenced signaling pathways, Phys. Biol., 13 (2016), 046004. https://doi.org/10.1088/1478-3975/13/4/046004 doi: 10.1088/1478-3975/13/4/046004
|
| [17] |
A. Sayyidmousavi, K. Rohlf, Stochastic simulations of the Schnakenberg model with spatial inhomogeneities using reactive multiparticle collision dynamics, AIMS Math., 4 (2019), 1805–1823. https://doi.org/10.3934/math.2019.6.1805 doi: 10.3934/math.2019.6.1805
|
| [18] |
A. Sayyidmousavi, K. Rohlf, S. Ilie, A hybrid method for micro-mesoscopic stochastic simulation of reaction-diffusion systems, Math. Biosci., 312 (2019), 23–32. https://doi.org/10.1016/j.mbs.2019.04.001 doi: 10.1016/j.mbs.2019.04.001
|
| [19] |
A. Sayyidmousavi, K. Rohlf, Reactive multi-particle collision dynamics with reactive boundary conditions, Phys. Biol., 15 (2018), 046007. https://doi.org/10.1088/1478-3975/aabc35 doi: 10.1088/1478-3975/aabc35
|
| [20] |
A. Malevanets, R. Kapral, Mesoscopic model for solvent dynamics, J. Chem. Phys., 110 (1999), 8605–8613. https://doi.org/10.1063/1.478857 doi: 10.1063/1.478857
|
| [21] |
S. Bedkihal, J. C. Kumaradas, K. Rohlf, Steady flow through a constricted cylinder by multiparticle collision dynamics, Biomech. Model. Mechanobiol., 12 (2013), 929–939. https://doi.org/10.1007/s10237-012-0454-z doi: 10.1007/s10237-012-0454-z
|
| [22] |
T. Akhter, K. Rohlf, Quantifying compressibility and slip in multiparticle collision (MPC) flow through a local constriction, Entropy, 16 (2014), 418–442. https://doi.org/10.3390/e16010418 doi: 10.3390/e16010418
|
| [23] |
S. Rabba, K. Rohlf, Pressure curves for compressible flows with slip through asymmetric local constrictions, IJANS, 3 (2018), 21–40. https://doi.org/10.1504/IJANS.2018.097324 doi: 10.1504/IJANS.2018.097324
|
| [24] | K. Rohlf, Theoretical studies on blood flow in small vessels, Ph.D. Thesis, University of Waterloo, 2002. |
| [25] |
T. Murata, Theory of non-Newtonian viscosity of blood at low shear rate — Effect of rouleaux, Biorheol., 13 (1976), 287–296. https://doi.org/10.3233/BIR-1976-13504 doi: 10.3233/BIR-1976-13504
|
| [26] |
T. Murata, T. W. Secomb, Effects of shear rate on rouleau formation in simple shear flow, Biorheol., 25 (1988), 113–122. https://doi.org/10.3233/BIR-1988-251-218 doi: 10.3233/BIR-1988-251-218
|
| [27] | J. Chen, Z. Huang, Analytical model for effects of shear rate on rouleau size and blood viscosity, Biophys. Chem., 58 (1996), 273–279. https://doi.org/10.1016/0301-4622(95)00105-0 |
| [28] | D. Barrows, Accurate and efficient stochastic simulation of reaction-diffusion biochemical networks, Ph.D. Thesis, Toronto Metropolitan University, 2025. |
| [29] | P. Kunwar, Stochastic simulations for extended Becker-Döring models applied to self-assembly processes, Ph.D. Thesis, Toronto Metropolitan University, 2025. |
| [30] | B. Szala-Mendyk, A. Drajkowska, A. Molski, Modified Smoluchowski rate equations for aggregation and fragmentation in finite systems, J. Phys. Chem. B., 127 (2023), 6154–6162. https://doi.org/10.1021/acs.jpcb.3c02884 |
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Pradeep Kunwar, Katrin Rohlf. Mean first assembly times in coagulation-fragmentation systems: Theoretical, stochastic, and approximate approaches[J]. AIMS Mathematics, 2026, 11(9): 31476-31515. doi: 10.3934/math.20261242
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