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A modified Morris–Lecar model for interacting ion channels
ARTICLE IN PRESS
Neurocomputing 64 (2005) 543–545 www.elsevier.com/locate/neucom
Letters
A modified Morris–Lecar model for interacting ion channels Ji-Huan Hea,b,, Xu-Hong Wua a
Institute of Fibrous Soft Matter Physics, College of Science, Donghua University, 1882 Yan’an Xilu Road, P.O. Box 471, Shanghai 200051, China b Key Lab of Textile Technology, Ministry of Education, Shanghai, China Received 22 November 2004; received in revised form 3 December 2004; accepted 3 December 2004 Communicated by R.W. Newcomb Available online 19 January 2005
Abstract A modified Morris–Lecar model is suggested taking into account the size effect of the interacting ionic channels. r 2004 Elsevier B.V. All rights reserved. Keywords: Modified Morris–Lecar model; Resistance of ionic channel
Morris and Lecar proposed a model to explain the electrical behavior of the barnacle muscle fiber, the model can be expressed in the following forms [5,1] C
dV ¼ ¯gCa m1 ðV V Ca Þ g¯ K wðV V k Þ gL ðV V L Þ þ I app ; dt
dw fðw1 wÞ ¼ ; dt t
(1)
(2)
Corresponding author. Institute of Fibrous Soft Matter Physics, College of Science, Donghua University, P.O. Box 471, Shanghai 200051, China. Tel.: +86 21 62379917; fax: +86 21 62373137. E-mail address: [email protected] (J.-H. He).
0925-2312/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2004.12.003
ARTICLE IN PRESS J.-H. He, X.-H. Wu / Neurocomputing 64 (2005) 543–545
544
where g¯ Ca ; g¯ K ; gL are normalization constants that determine the maximum possible conductances for, respectively, Ca2þ ; Kþ ; and the leak. Note that the constants g¯ Ca ; g¯ K ; and gL have different values for channels of different radii, leading to inconvenience when being applied to the Morris–Lecar model. If we assume that the total conductance is the result of the activation of many channels, the conductance gCa can be defined as the product of the maximum possible conductance ð¯gCa Þ and the fraction of open channels ðm1 Þ [1], gCa ¼ g¯ Ca m1 ;
(3)
The basic underlying perception for conductance of ionic channels, however, has remained elusive. We proposed an allometric scaling law in the form [3] gi r a ;
(4)
where r is the radius of the ionic channel, a is the scaling exponent. When a tends to 2, and it becomes the metal-like conductor; when a ¼ 0; it turns out to be an electric insulator. For surface charge conduction, e.g. electrospinning [4,2,6], a ¼ 1; i.e., the conductance for surface convection scales as gs ¼ 1=Rs r; and the surface convection current is I s ¼ 2prsu; where s is the surface density of the charge. Taking into account the size effect of the muscle fiber, the current balance equation is modified as a a dV r r ¼ G Ca m1 ðV V Ca Þ C G K wðV V k Þ dt r0 r0 a r G L ðV V L Þ þ I app ; ð5Þ r0 where the terms GCa ; GK ; and GL represent maximal conductances when the radius of the fiber is r0 : Generally r0 is chosen as the radius of a certain giant barnacle muscle fiber. We have proposed a modified Morris–Lecar model, Eq. (5). The idea can be readily extended to other famous models in biology, such as the Hodgkin–Huxley model, FitzHugh–Nagumo model, and others. Of course the authors understand that no matter how rigorous, some experimental verification is needed to validate the model.
Acknowledgment The authors thank the unknown reviewers for their careful reading and helpful comments. The work is supported by Grant 10372021 from National Natural Science Foundation of China.
ARTICLE IN PRESS J.-H. He, X.-H. Wu / Neurocomputing 64 (2005) 543–545
545
References [1] C.P. Fall, E.S. Marland, J.M. Wagner, J.J. Tyson, Computational Cell Biology, Springer, New York, 2002. [2] J.H. He, Int. J. Nonlinear Sci. Numer. Simul. 5 (3) (2004) 263–274. [3] J.H. He, Y.Q. Wan, J.Y. Yu, Int. J. Nonlinear Sci. Numer. Simul. 5 (3) (2004) 243–252. [4] J.H. He, Y.Q. Wan, J.Y. Yu, Int. J. Nonlinear Sci. Numer. Simul. 5 (3) (2004) 253–261. [5] C. Morris, H. Lecar, Voltage oscillations in the barnacle giant muscle, Biophys. J. 35 (1981) 193–213. [6] Y.Q. Wan, Q. Guo, N. Pan, Int. J. Nonlinear Sci. Numer. Simul. 5 (1) (2004) 5–8.
Neurocomputing 64 (2005) 543–545 www.elsevier.com/locate/neucom
Letters
A modified Morris–Lecar model for interacting ion channels Ji-Huan Hea,b,, Xu-Hong Wua a
Institute of Fibrous Soft Matter Physics, College of Science, Donghua University, 1882 Yan’an Xilu Road, P.O. Box 471, Shanghai 200051, China b Key Lab of Textile Technology, Ministry of Education, Shanghai, China Received 22 November 2004; received in revised form 3 December 2004; accepted 3 December 2004 Communicated by R.W. Newcomb Available online 19 January 2005
Abstract A modified Morris–Lecar model is suggested taking into account the size effect of the interacting ionic channels. r 2004 Elsevier B.V. All rights reserved. Keywords: Modified Morris–Lecar model; Resistance of ionic channel
Morris and Lecar proposed a model to explain the electrical behavior of the barnacle muscle fiber, the model can be expressed in the following forms [5,1] C
dV ¼ ¯gCa m1 ðV V Ca Þ g¯ K wðV V k Þ gL ðV V L Þ þ I app ; dt
dw fðw1 wÞ ¼ ; dt t
(1)
(2)
Corresponding author. Institute of Fibrous Soft Matter Physics, College of Science, Donghua University, P.O. Box 471, Shanghai 200051, China. Tel.: +86 21 62379917; fax: +86 21 62373137. E-mail address: [email protected] (J.-H. He).
0925-2312/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2004.12.003
ARTICLE IN PRESS J.-H. He, X.-H. Wu / Neurocomputing 64 (2005) 543–545
544
where g¯ Ca ; g¯ K ; gL are normalization constants that determine the maximum possible conductances for, respectively, Ca2þ ; Kþ ; and the leak. Note that the constants g¯ Ca ; g¯ K ; and gL have different values for channels of different radii, leading to inconvenience when being applied to the Morris–Lecar model. If we assume that the total conductance is the result of the activation of many channels, the conductance gCa can be defined as the product of the maximum possible conductance ð¯gCa Þ and the fraction of open channels ðm1 Þ [1], gCa ¼ g¯ Ca m1 ;
(3)
The basic underlying perception for conductance of ionic channels, however, has remained elusive. We proposed an allometric scaling law in the form [3] gi r a ;
(4)
where r is the radius of the ionic channel, a is the scaling exponent. When a tends to 2, and it becomes the metal-like conductor; when a ¼ 0; it turns out to be an electric insulator. For surface charge conduction, e.g. electrospinning [4,2,6], a ¼ 1; i.e., the conductance for surface convection scales as gs ¼ 1=Rs r; and the surface convection current is I s ¼ 2prsu; where s is the surface density of the charge. Taking into account the size effect of the muscle fiber, the current balance equation is modified as a a dV r r ¼ G Ca m1 ðV V Ca Þ C G K wðV V k Þ dt r0 r0 a r G L ðV V L Þ þ I app ; ð5Þ r0 where the terms GCa ; GK ; and GL represent maximal conductances when the radius of the fiber is r0 : Generally r0 is chosen as the radius of a certain giant barnacle muscle fiber. We have proposed a modified Morris–Lecar model, Eq. (5). The idea can be readily extended to other famous models in biology, such as the Hodgkin–Huxley model, FitzHugh–Nagumo model, and others. Of course the authors understand that no matter how rigorous, some experimental verification is needed to validate the model.
Acknowledgment The authors thank the unknown reviewers for their careful reading and helpful comments. The work is supported by Grant 10372021 from National Natural Science Foundation of China.
ARTICLE IN PRESS J.-H. He, X.-H. Wu / Neurocomputing 64 (2005) 543–545
545
References [1] C.P. Fall, E.S. Marland, J.M. Wagner, J.J. Tyson, Computational Cell Biology, Springer, New York, 2002. [2] J.H. He, Int. J. Nonlinear Sci. Numer. Simul. 5 (3) (2004) 263–274. [3] J.H. He, Y.Q. Wan, J.Y. Yu, Int. J. Nonlinear Sci. Numer. Simul. 5 (3) (2004) 243–252. [4] J.H. He, Y.Q. Wan, J.Y. Yu, Int. J. Nonlinear Sci. Numer. Simul. 5 (3) (2004) 253–261. [5] C. Morris, H. Lecar, Voltage oscillations in the barnacle giant muscle, Biophys. J. 35 (1981) 193–213. [6] Y.Q. Wan, Q. Guo, N. Pan, Int. J. Nonlinear Sci. Numer. Simul. 5 (1) (2004) 5–8.