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BME 322 - Probability and Applications for BME Project 1: Probabilistic Model of lon Channel Behavior Due Sunday, 27 March Objective: The objectives of this assignment are: (1) to create a probabilistic model of ion channel behavior and compare the summed behavior of a large number of individual ion channels with the whole-cell predictions of the Hodgkin- Huxley squid giant axon model; and (2) to assess the model's performance based on how well it achieves a set of design requirements. Background: As discussed last semester in BME 33400, in the 1950s A.L Hodgkin and A.F. Huxley characterized the electrical behavior of a squid axon using an elegant series of experimental and analytical techniques. The result was the first computational model describing the formation of an action potential. This model describes the cell membrane as an electrical circuit, with a capacitance Cm representing charge separation of the lipid bilayer of the membrane, and conductances" (the inverse of resistance) describing the movement of ions across the membrane as a function of time and voltage. Ion channels had not yet been discovered - so Hodgkin and Huxley had to hypothesize a theoretical process that allowed sodium (Na") and potassium (K) ions to cross the membrane. They described these processes as "gating particles that enabled or impeded the movement of ions across the membrane. For Na', they hypothesized two types of gating particles, which they called "m" (an activation particle) and an inactivation particle). For K, they proposed a single type of gating particle. "n" (an activation particle). They also described a small "leak current to account for current leaking around their recording electrodes. inside outside The resistors in the circuit represent time and voltage-dependent) conductance of ions, the capacitor represents membrane capacitance, and the batteries represent the electrotonic force resulting from having unequal concentrations of ions inside and outside the cell. For example, under equilibrium conditions, Nat is at a much higher concentration outside the cell than inside the cell. Therefore, when ion channels open, this gradient means Na is likely to flow inward. K at rest is at higher concentration inside the cell, so its gradient would naturally lead it to flow outward.
Deterministic Whole-Cell Model Arises from Probabilistic Behavior of lon Channels The voltage-dependent steady-state activation and inactivation functions (mus. hos, nsa; see equations below) describe the probability for a given voltage, that an ion channel is in the activated or inactivated state. Using the frequency description of probability, we can also view this as the expected proportion of ion channels that are in the relevant state, when the total number of ion channels is some large number. Thus, these functions can be thought to characterize the lumped, summed or averaged behavior of a large number of ion channels. In response to a step of voltage, we would expect whole-cell current - the net effect of the sum of all the individual ion channel currents in the cell-to rise at some rate determined by its time-constant and eventually settle at some steady- state value The behavior of an individual ion channel, on the other hand, is quite different from this. At a given voltage, a single ion channel flickers on and off. Its open- channel probability determines the likelihood that it is open at each instant; thus, the proportion of time it is open (when voltage is held fixed) should approach this probability over a large number of trials. However, the individual time trajectory may vary widely from one ion channel to the next, or even from one experiment to the next on the same ion channel: பபப Two son channels with similar open probability but different time-trajectories Summing together a large number of individual ion channel currents produces the whole-cell current. Experimental evidence (Hille 2001, summarized from Levinson and Meves 1975: Strichartz et al 1979, and Conti et al 1975) suggests there are on the order of 200 to 500 sodium ion channels and about 30 potassium ion channels per um? of membrane in the squid giant axon.
Hodgkin-Huxley Equation Set The differential equation describing the membrane potential (Vm) in the squid axon membrane model is: dvm - 1 + x +he - Lin] dt CM Here, Iva, Ik, and I represent membrane sodium, potassium and nonspecified leak current and Cu represents membrane capacitance. Unless otherwise noted the following units apply: voltage in mV, current density in A/cm², conductance per unit area in mS/cm2, time in msec and specific capacitance in uF/cm. There are three other differential equations in the Hodgkin-Huxley model. These describe the (voltage-dependent) rates of change of the activation and inactivation state variables (m, h, n). These equations are derived from the process of transitioning from the state of not allowing current to flow to the state of "allowing current to flow": a(V) 1-0 BV) Sodium Current Equations: I gm'.-E. a. (25-V)/10 (-1 +exp((25-V)/10)) B. = 4,00 exp(-(V)/18.0) a, = 0.07 exp(-(V)/20) 1.0 B. (1 + exp{(30-V)/10) dh =a. (1-1)-B,H dt don dr a (1-m-B.. a h- a. + 1.0 a. +. 1.0 a,+B.
Hodgkin-Huxley Equation Set The differential equation describing the membrane potential (Vm) in the squid axon membrane model is: dvm - 1 + x +he - Lin] dt CM Here, Iva, Ik, and I represent membrane sodium, potassium and nonspecified leak current and Cu represents membrane capacitance. Unless otherwise noted the following units apply: voltage in mV, current density in A/cm², conductance per unit area in mS/cm2, time in msec and specific capacitance in uF/cm. There are three other differential equations in the Hodgkin-Huxley model. These describe the (voltage-dependent) rates of change of the activation and inactivation state variables (m, h, n). These equations are derived from the process of transitioning from the state of not allowing current to flow to the state of "allowing current to flow": a(V) 1-0 BV) Sodium Current Equations: I gm'.-E. a. (25-V)/10 (-1 +exp((25-V)/10)) B. = 4,00 exp(-(V)/18.0) a, = 0.07 exp(-(V)/20) 1.0 B. (1 + exp{(30-V)/10) dh =a. (1-1)-B,H dt don dr a (1-m-B.. a h- a. + 1.0 a. +. 1.0 a,+B.
Potassium Current Equations: k = n(V.-EX) B. = 0.125 exp(-V /80.0) (10-1)/100 a. (-1+exp(10-V)/10)) dn =a. (1-1)-B. di Q 4. + 1.0 4. B. Leakage Current: 1. = g.V.-E.) Model Constants: CM = 1.0 uF/cm Ex = -12.0 mv = 120 mS/cm2 9 = 0.3 mS/cm2 Eno = 115 mV Ex = 10.6 mV Q = 36 mS/cm lasin = 200 xA/cm2 for 0.05 msec = Initial Conditions: V(t) = 0 mv (V., to) = has(Omv) m(V. 1) = m.:(0 mV) n(Vota) = nss(0 mV) Note that these equations have all been formulated to give a "resting membrane potential of 0 mV. You may sometimes see slightly different equations used that result in a resting membrane potential closer to -60 mV. This latter formulation is consistent with modern conventions, which consider the outside of the cell to be "ground" (0 mV) and the voltage of the cell to be the potential inside relative to the potential outside.