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Radhakrishnan K, Halász A, Vlachos D, Edwards JS. Quantitative understanding of cell signaling: the importance of membrane organization. Curr Opin Biotechnol 2010; 21:677-82. [PMID: 20829029 DOI: 10.1016/j.copbio.2010.08.006] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2010] [Accepted: 08/09/2010] [Indexed: 12/13/2022]
Abstract
Systems biology modeling of signal transduction pathways traditionally employs ordinary differential equations, deterministic models based on the assumptions of spatial homogeneity. However, this can be a poor approximation for certain aspects of signal transduction, especially its initial steps: the cell membrane exhibits significant spatial organization, with diffusion rates approximately two orders of magnitude slower than those in the cytosol. Thus, to unravel the complexities of signaling pathways, quantitative models must consider spatial organization as an important feature of cell signaling. Furthermore, spatial separation limits the number of molecules that can physically interact, requiring stochastic simulation methods that account for individual molecules. Herein, we discuss the need for mathematical models and experiments that appreciate the importance of spatial organization in the membrane.
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Affiliation(s)
- Krishnan Radhakrishnan
- Department of Pathology and Cancer Center, University of New Mexico Health Sciences Center, Albuquerque, NM 87131, USA
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Grima R, Schnell S. How Reaction Kinetics with Time-Dependent Rate Coefficients Differs from Generalized Mass Action. Chemphyschem 2006; 7:1422-4. [PMID: 16755640 DOI: 10.1002/cphc.200600081] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Ramon Grima
- Indiana University School of Informatics and Biocomplexity Institute, Eigenmann Hall 906, 1900 East Tenth Street, Bloomington, IN 47406, USA
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Mayawala K, Vlachos DG, Edwards JS. Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations. Biophys Chem 2006; 121:194-208. [PMID: 16504372 DOI: 10.1016/j.bpc.2006.01.008] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2006] [Accepted: 01/19/2006] [Indexed: 12/17/2022]
Abstract
Bimolecular reactions in the plasma membrane, such as receptor dimerization, are a key signaling step for many signaling systems. For receptors to dimerize, they must first diffuse until a collision happens, upon which a dimerization reaction may occur. Therefore, study of the dynamics of cell signaling on the membrane may require the use of a spatial modeling framework. Despite the availability of spatial simulation methods, e.g., stochastic spatial Monte Carlo (MC) simulation and partial differential equation (PDE) based approaches, many biological models invoke well-mixed assumptions without completely evaluating the importance of spatial organization. Whether one is to utilize a spatial or non-spatial simulation framework is therefore an important decision. In order to evaluate the importance of spatial effects a priori, i.e., without performing simulations, we have assessed the applicability of a dimensionless number, known as second Damköhler number (Da), defined here as the ratio of time scales of collision and reaction, for 2-dimensional bimolecular reactions. Our study shows that dimerization reactions in the plasma membrane with Da approximately >0.1 (tested in the receptor density range of 10(2)-10(5)/microm(2)) require spatial modeling. We also evaluated the effective reaction rate constants of MC and simple deterministic PDEs. Our simulations show that the effective reaction rate constant decreases with time due to time dependent changes in the spatial distribution of receptors. As a result, the effective reaction rate constant of simple PDEs can differ from that of MC by up to two orders of magnitude. Furthermore, we show that the fluctuations in the number of copies of signaling proteins (noise) may also depend on the diffusion properties of the system. Finally, we used the spatial MC model to explore the effect of plasma membrane heterogeneities, such as receptor localization and reduced diffusivity, on the dimerization rate. Interestingly, our simulations show that localization of epidermal growth factor receptor (EGFR) can cause the diffusion limited dimerization rate to be up to two orders of magnitude higher at higher average receptor densities reported for cancer cells, as compared to a normal cell.
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Affiliation(s)
- Kapil Mayawala
- Department of Chemical Engineering, 150 Academy Street, University of Delaware, Newark, DE 19716, USA
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Abad E. First-passage method for the study of the efficiency of a two-channel reaction on a lattice. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:021107. [PMID: 16196546 DOI: 10.1103/physreve.72.021107] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2005] [Indexed: 05/04/2023]
Abstract
We study the efficiency of a two-channel reaction between two walkers on a finite one-dimensional periodic lattice. The walkers perform a combination of synchronous and asynchronous jumps on the lattice and react instantaneously when they meet at the same site (first channel) or upon position exchange (second channel). We develop a method based on a conditional first-passage problem to obtain exact results for the mean number of time steps needed for the reaction to take place as well as for higher order moments. Previous results obtained in the framework of a difference equation approach are fully confirmed, including the existence of a parity effect. For even lattices the maximum efficiency corresponds to a mixture of synchronous events and a small amount of asynchronous events, while for odd lattices the reaction time is minimized by a purely synchronous process. We provide an intuitive explanation for this behavior. In addition, we give explicit expressions for the variance of the reaction time. The latter displays a similar even-odd behavior, suggesting that the parity effect extends to higher order moments.
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Affiliation(s)
- E Abad
- Centre for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, CP 231, 1050 Bruxelles, Belgium.
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Deloubrière O, Hilhorst HJ, Täuber UC. Multispecies pair annihilation reactions. PHYSICAL REVIEW LETTERS 2002; 89:250601. [PMID: 12484872 DOI: 10.1103/physrevlett.89.250601] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2002] [Indexed: 05/24/2023]
Abstract
We consider diffusion-limited reactions A(i)+A(j)--> (1< or =i<j< or =q) in d space dimensions. For q>2 and d> or =2, we argue that the asymptotic density decay for such mutual annihilation processes with equal rates and initial densities is the same as for single-species pair annihilation A+A-->. In d=1, however, particle segregation occurs for all q< infinity. The total density decays according to a q dependent power law, rho(t) approximately t(-alpha(q)). Within a simplified version of the model alpha(q)=(q-1)/2q can be determined exactly. Our findings are supported through Monte Carlo simulations.
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Affiliation(s)
- Olivier Deloubrière
- Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg 24061-0435, USA
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Bray AJ, Blythe RA. Exact asymptotics for one-dimensional diffusion with mobile traps. PHYSICAL REVIEW LETTERS 2002; 89:150601. [PMID: 12365977 DOI: 10.1103/physrevlett.89.150601] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/2002] [Indexed: 05/23/2023]
Abstract
We consider a diffusing particle, with diffusion constant D', moving in one dimension in an infinite sea of noninteracting mobile traps with diffusion constant D and density rho. We show that the asymptotic behavior of the survival probability, P(t), satisfies lim([-ln(P(t)]/sqrt[rho2Dt]=4/sqrt[pi], independent of D'. The result comes from obtaining upper and lower bounds on P(t), and showing that they coincide asymptotically. We also obtain exact results for P(t) to first order in D(')/D for an arbitrary finite number of traps.
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Affiliation(s)
- Alan J Bray
- Department of Physics and Astronomy, University of Manchester, United Kingdom
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Mehra V, Grassberger P. Trapping reaction with mobile traps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:050101. [PMID: 12059510 DOI: 10.1103/physreve.65.050101] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2001] [Indexed: 05/23/2023]
Abstract
We present the Monte Carlo results for the two-species trapping reaction A+B-->B with diffusing A and B on lattices in one, two, and three dimensions. We use an algorithm that permits one to simulate the survival probabilities of A particles down to <10(-30) with high accuracy. The results for the survival probability agree much better with the exact asymptotic predictions of Bramson and Lebowitz [Phys. Rev. Lett. 61, 2397 (1988)] than with the heuristics of Kang and Redner [J. Phys. A 17, L451 (1984)]. But there are very large deviations from either, which show that even these simulations are far from asymptotia. This is supported by the rms displacement of A particles, which clearly shows that the asymptotic regime has not been reached, at least for d=2 and d=3.
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Affiliation(s)
- Vishal Mehra
- John-von-Neumann Institute for Computing, Forschungszentrum Jülich, D-52425 Jülich, Germany
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Adrover A, Giordano F. Shortcut Method for Lumping Diffusion−Reaction Kinetics in Lamellar Systems. Ind Eng Chem Res 1999. [DOI: 10.1021/ie990353a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Alessandra Adrover
- Dipartimento di Ingegneria Chimica, Universitá di Roma “La Sapienza”, via Eudossiana 18, 00184 Roma, Italy
| | - Francesca Giordano
- Dipartimento di Ingegneria Chimica, Universitá di Roma “La Sapienza”, via Eudossiana 18, 00184 Roma, Italy
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Reigada R, Sagués F, Sokolov IM, Sancho JM, Blumen A. Kinetics of the A+B→0 reaction under steady and turbulent flows. J Chem Phys 1996. [DOI: 10.1063/1.472896] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Oshanin G, Sokolov IM, Argyrakis P, Blumen A. Fluctuation‐dominated A+B→0 kinetics under short‐ranged interparticle interactions. J Chem Phys 1996. [DOI: 10.1063/1.472466] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Reigada R, Sagués F, Sokolov IM, Sancho JM, Blumen A. Spatial correlations and cross sections of clusters in the A+B-->0 reaction. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:3167-3172. [PMID: 9964624 DOI: 10.1103/physreve.53.3167] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Abramson G, Rodriguez MA, Wio HS. Analytic approach to the space-time kinetics of annihilation reactions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:4319-4326. [PMID: 9962513 DOI: 10.1103/physreve.50.4319] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Sokolov IM, Blumen A. Kinetics in coagulation-annihilation processes. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:2335-2338. [PMID: 9962242 DOI: 10.1103/physreve.50.2335] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Mai J, Kuzovkov VN. Stochastic model for complex surface-reaction systems with application to NH3 formation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:1700-1709. [PMID: 9960781 DOI: 10.1103/physreve.48.1700] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Kuzovkov V, Kotomin E. Dynamic particle aggregation in the bimolecularA+B→0 reaction. J Chem Phys 1993. [DOI: 10.1063/1.464468] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Argyrakis P, Kopelman R. Nonclassical A+B-->0 batch reactions: Effect of mobility on rate, order, aggregation and segregation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:3757-3760. [PMID: 9960437 DOI: 10.1103/physreve.47.3757] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Kroon R, Fleurent H, Sprik R. Diffusion-limited exciton fusion reaction in one-dimensional tetramethylammonium manganese trichloride (TMMC). PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:2462-2472. [PMID: 9960278 DOI: 10.1103/physreve.47.2462] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Larralde H, Araujo M, Havlin S, Stanley HE. Reaction front for A+B-->C diffusion-reaction systems with initially separated reactants. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:855-859. [PMID: 9908186 DOI: 10.1103/physreva.46.855] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Araujo M, Havlin S, Larralde H, Stanley HE. "Random-force-dominated" reaction kinetics: Reactants moving under random forces. PHYSICAL REVIEW LETTERS 1992; 68:1791-1794. [PMID: 10045221 DOI: 10.1103/physrevlett.68.1791] [Citation(s) in RCA: 26] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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