A new double-chamber model of ion channels. Beyond the Hodgkin and Huxley model

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This paper proposes a new double-chamber model (DCM) of ion channels. The model ion channel consists of a series of three pores alternating with two chambers. The chambers are net negatively charged. The chamber's electric charge originates from
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  CELLULAR & MOLECULAR BIOLOGY LETTERS Volume 8, (2003) pp 749 – 775 http://www.cmbl.org.pl   Received 21 February 2003 Accepted 9 June 2003 Abbreviations used: HHM - Hodgkin and Huxley Model; DCM - Double-Chamber Model A NEW DOUBLE-CHAMBER MODEL OF ION CHANNELS. BEYOND THE HODGKIN AND HUXLEY MODEL KRZYSZTOF DO  OWY Department of Biophysics, Warsaw Agricultural University - SGGW, Rakowiecka 26/30, 02-528 Warszawa, Poland  Abstract : This paper proposes a new double-chamber model (DCM) of ion channels. The model ion channel consists of a series of three pores alternating with two chambers. The chambers are net negatively charged. The chamber’s electric charge srcinates from dissociated amino acid side chains and is pH dependent. The chamber’s net negative charge is compensated by cations present inside the chamber and in a diffuse electric layer outside the chamber. The  pore’s permeability is constant independent of time. One pore of the sodium channel and one of the potassium channel is a voltage-sensing pore. Due to the channel's structure, ions flow through the pores and chambers in a time-dependent manner. The model reproduces experimental voltage clamp and action potential data. The current flowing through a single sodium channel is less then one femtoampere. The DCM is considerably simpler then the Hodgkin and Huxley model (HHM) used to describe the electrophysiological properties of an axon. Unlike the HHM, the DCM can explain refractoriness, anode break excitation, accommodation and the effect of pH and temperature on the channels without additional parameters. In the DCM, the axon membrane shows repetitive activity depending on the channel density, sodium to potassium channel ratio and external potassium concentration. In the DCM, the action potential starts from ‘hot spot areas’ of higher channel densities and a higher sodium to  potassium channel ratio, and then propagates through the whole axon.   Key Words : Hodgkin and Huxley Model, Sodium Channel, Potassium Channel, Voltage Clamp, Action Potential, Femtoampere Channel INTRODUCTION The formation of an action potential in the axon of a nerve cell has been studied for a very long time – for a review, see e.g. [1, 2]. Experiments performed by  CELL. MOL. BIOL. LETT. Vol. 8. No. 3. 2003 750 Hodgkin and Huxley [3-5] explained the sequence of events occurring during an action potential. In short, during the depolarization of the membrane above the threshold potential value, sodium ions flow into the cell and the membrane  potential reaches a positive value. Within a short time (around 2 ms at 4 ! C) the number of sodium ions flowing into the cell decreases. Next, potassium ions flow out from the cell and the potential returns to the resting potential value. Hodgkin and Huxley [6] formulated a theory of how sodium and potassium channels operate. In short, each channel has four gating particles. Potassium channel particles, denoted n , move from a closed to an open state, and potassium channel conductance is described by the function n 4 . The function n  is described  by first order kinetics. Sodium conductance is described by the function m 3 h . Functions m  and n  rise with time from minimal to maximal value and function h  decreases with time. All three ( m , n  and h ) are also functions of membrane  potential. The HHM is fitted with a large number of parameters to adjust conductance functions to voltage clamp empirical data. The potassium conductance function has 10 parameters, which cannot be independently measured, and 3 arbitrary functions of membrane potential. In the more complicated sodium conductance function, twice as many parameters and functions of membrane potential are used. The HHM is an extremely successful theory, describing not only action potential  but also other well-known neurophysiological properties of the axon such as threshold, refractoriness, strength-duration relationship, anode break excitation, repetitive activity and subthreshold oscillations – for a review, see [2, 7, 8]. The HHM introduced a concept of separate pathways for sodium and potassium, which led to the discovery of ion channels and the patch clamp technique [9]. The HHM has become the fundament of modern biophysical teaching to the extent that any criticism of it is almost treated like blasphemy. However, in the last 50 years, some weak points of the HHM have been found. The HHM assumes that the gating particle movement is responsible for the change of channel conductance, but experiments show that this is not the case. Gating movement is completed within 0.1 ms, while the process of action potential formation takes at least 2 ms [2, 8]. According to the HHM m 3 h  model, the deactivation time constant should be three times faster (one of the m  particles should move from the permissive to the nonpermissive state) than the activation time constant. This is also not the case [2, 8]. The physical model behind the HHM is refuted and we are left with the mathematical part of the model, which Hille [1] describes as “a smoke screen of the possible physical meaning of the functions and parameters used in the model allowed easier acceptance of the HHM by biologists”. Another weak point of the HHM is that it does not predict the effects of such  parameters as temperature, viscosity or the pH of the medium. The HHM may be supplemented with three ad hoc  functions of temperature that enable the description of the action potential at different temperatures. However, even  CELLULAR & MOLECULAR BIOLOGY LETTERS 751 supplemented with a temperature function, it cannot explain the effect of temperature on accommodation [10, 11]. The discovery of ion channels has not resolved the difficulties which face the HHM but created new ones. The HHM requires that the whole membrane  permeability to sodium and potassium ions should be described by a continuous function. If one assumes that in the axon membrane there are many channels, uniformly distributed, each conducting small amount of ions, then the HHM could be saved. However, if there are very few ion channels in a cell membrane or only a few out of many open during action potential, then such properties as refractoriness, strength-duration relationship, repetitive activity and accommodation cannot be readily explained. The purpose of this theoretical paper is to show that, instead of a complex mathematical model of sodium and potassium channels, one can use a simple  physical model of ion channel, which consist of three resistances and two capacitances. None of the model’s electric component is time-dependent. However, the set of these components can describe time-dependent properties of a nerve cell during voltage clamp and an action potential. It is assumed that in the membrane, there are numerous ion channels, each conducting small current (beyond the sensitivity of the patch clamp technique) and in that sense it continues the tradition of the HHM. THE MODEL Model channel structure  The ion channel in the model consists of a series of three pores alternating with two chambers (Fig. 1). The permeability of a pore is dependent on neither time nor membrane potential. The exception is one pore of the channel, which has a voltage-sensing gate. The movement of ions in the pore is described by classic diffusion and the concentration of ions in the chamber by a classic adsorption function. A model lacking one of the constituents presented in Fig. 1 cannot describe the axon properties. Chambers are net negatively charged The chamber’s electric charge srcinates from dissociated amino acid side chains. The amino acid composition of both the sodium [12] and potassium channels [13] is known (Tab. 1). The number of a chamber’s charges depends on the pH of the solution present inside the chamber, which is a function of the  pH  i  of the medium outside the chamber and the potential difference  ij . For the dissociation of both types AH ⇔ A -  + H + and BH + ⇔ B + H +    CELL. MOL. BIOL. LETT. Vol. 8. No. 3. 2003 752 Fig. 1. The double chamber model of the sodium ion channel. The numbers correspond to four media: external medium ‘0’; external chamber ‘1’; internal chamber ‘2’; and internal medium ‘3’. The media are denoted by subscripts, e.g. the current flowing through the pore between the external medium and the external chamber is denoted by  I  01 . Circles represent cations flowing through a channel. Squares represent negatively charged side chains of amino acids. Electric components and their physical meaning are shown. The sodium channel has a very wide 23Na pore. The sodium channel 12Na pore is the voltage-sensing one. The model also describes the potassium channel; however, the voltage-sensing pore is a 23K one and the very wide one is a 01K pore. Tab. 1. The number of dissociable amino acids on the external and internal part of the channel protein. D – aspartic acid, E – glutamic acid, K – lysine, R – arginine, C – cysteine, H – histidine, Y – tyrosine. The number of amino acids which are located within the lipid bilayer was arbitrarily halved and added to the external and internal part of protein. Amino acid pK Voltage-gated Na-channel hNab 2  [12] Voltage-gated K-channel mKv1.1 (MK1)*4 [13] external internal external internal D - Asp 3.86 32 69 24 72 E - Glu 4.07 32 110 40 128 K - Lys 8.95 25 107 20 56 R - Arg 12.48 19 63 16 100 C - Cys 8.00 17 20 8 24 H - His 6.10 6 13 8 28 Y - Tyr 10.07 25 33 12 60  –COOH 2.3 0 4 0 4  –NH 2 9.5 0 4 0 4  CELLULAR & MOLECULAR BIOLOGY LETTERS 753 The ratio of concentration of more to less negatively charged forms of the dissociable side chains of the amino acids of a dissociation constant  K   are given  by [ ][ ] [ ] [ ]  −∗== +−  RT  F  H  K  BH  B AH  A iji ϕ exp  (1) where the subscripts i,j  corresponds to either 0,1  or 2,3 . The change in the pH of the medium and the change in potential difference across the chamber changes the number of net charges of the chamber; i.e.  N  1Na ,  N  2Na ,  N  1K  ,  N  2K  . The maximum net charge of the chamber for pH=7 can be calculated assuming that the side chains of all the amino acids are dissociated. The lower limit of the chamber charge can be estimated by fitting the model to experimental data. Below the lower limit of the charge value, the data cannot be fitted to experimental data. For the sodium channel, the net charge of the external part of the channel molecule is –12 ≤  N  1Na ≤  –19, and on the cytoplasmic part –5 ≤  N  2Na ≤  –10, (the first value represents an estimated minimum, and the later the maximum of the charge). The net charges for the potassium channel molecule are –1 ≤  N  1K  ≤  –28 and –22 ≤  N  2K  ≤  –44 respectively. (The structure of the potassium channel studied  by X-ray analysis [14] reveals that the selectivity filter can contain two  potassium ions corresponding to  N  1K  , and more in a large water-filled cavity corresponding to  N  2K  ). Concentration of free counterions in the chamber  The chamber’s net negative charge is compensated by cations present inside the chamber and in a diffuse electric layer outside the chamber. Counterions present in the chamber are either adsorbed or free. In the case when the chamber’s charge is completely compensated by counterions, the concentration of free ions in the chamber and in the medium is equal. Thus, ion concentration inside the chambers is given by: 1101  N ncc  = , 2232  N ncc  =  (2) where n 1  and n 2  are the numbers of counterions present inside the chamber at a given moment, and c 0  and c 3  are the ion concentration in the medium and in the axoplasm, respectively. n 1  and n 2  are fractional quantities which represent the mean number of counterions present in the statistical chamber. Potential in chambers  If the counterions present inside a chamber do not compensate the net charge of the chamber, then a classic diffuse double layer potential outside the channel molecule is formed: ( ) 011101  ϕϕ  +−=  d  C q N n , ( ) 322223  ϕϕ  +−=  d  C q N n  (3)
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