Lecture 7 Notes, Thursday 15 September 2005

  • Fall et al., Chap. 2, pp. 27-43
  • Review Hille, Fig. 1.6 (p. 19)
  • Homework due: Thur 22 Sept 2005

    • Today we will add a new "active" current to the parallel conductance model of the cell membrane. The new current models a voltage-dependent calcium channel with instantaneous activation (and no "inactivation", which is a principle we will deal with soon). To describe voltage-dependent activation we will introduce gating variables that are responsible for opening the ion channels. We will analyze the electrical behavior of the model as we vary the applied current using IV curves, phase portraits, and ultimately a bifurcation diagram.

    Membrane with active calcium current: a bistable cell model (bistable.ode)

    ΣIM - Iapp = 0

    IC + IK + ICa - Iapp = 0

    CV′ + gK(V - EK) + gCa(V)(V - ECa) - Iapp = 0

    Looking in more detail at ICa,

    We will define a gating variable m for activation or opening the calcium channel, which at steady state is: mss(V). Multiplying the gating variable and maximum open conductance gCa (note this should have a 'hat' over g, but I use bold here instead because of HTML limitations) gives: gCa(V) = gCamss(V).

    So the voltage-dependent calcium current is: ICa = gCamss(V)(V - ECa).

    Thus the full current-balance equation,

    CV′ + gK(V - EK) + gCamss(V)(V - ECa) - Iapp = 0.

    We will further describe the voltage-dependence of activation (at steady state) using the expression:

    mss(V) = 0.5*[1 + TANH((V - VΘ)/σ)],

    which is a sigmoidal function that rises with half-maximal voltage VΘ and slope factor σ. Fall et al. describe how and why this particular function is used on p. 30.

    Go here for a reveiw of hyperbolic functions.

    We will plot the IV curve for the full membrane current, and then translate the IV curve into a phase diagram (V′ vs. VM), which we can analyze based on its fixed points. This system, we will see, is bistable, and the steady state of VM will depend on Iapp.

    We will introduce the heavyside step function H(t) in our work with the bistable model, which allows us to deliver current stimuli of fixed amplitude and duration: H(t) = 0 for t <= 0, H(t) = 1 for t > 0.

    You can download two versions of the model. The first (bistable.ode) utilizes a simple constant Iapp, whereas the second (bistable2.ode) has multiple heavyside step functions within the expression for Iapp.

    We will explore the role of Iapp in changing the behavior of the model, in particular, its role in determining the stability of the fixed points. We will show how Iapp acts as a bifurcation parameter and we will study the cell membrane behavior according to a bifurcation diagram. You may find it helpful to have access to the image of the bifurcation diagram on which to take notes. For the really aggressive student, try to generate the curve yourself using the AUTO function within XPPAUT. Label and analyze the diagram according to our discussions in class, be sure to find the stable and unstable steady states, as well as the values of Iapp that cause critical changes in the phase diagram and cell membrane behavior.

    Click to obtain a full-size PDF copy of this bifurcation diagram
    Homework (Due Thur 22 Sept 2005) answer key for Homework #4

    (1) Regarding space-charge neutrality, we showed that only a very few uncompensated charges are needed to produce large electrical potentials across the membrane. If membrane capacitance is 1 μF cm-2 (i.e., 10-6 uncompensated coulombs of charge on each side of a 1 cm2 patch of membrane are required to produce 1 V of potential difference), and the concentration of ions inside and outside the cell are about 0.1 M, calculate the fraction of uncompensated ions on each side of the membrane that are required to produce 100 mV for each of the following :

  • across a 1 cm2 patch of membrane (associate with 1 cm3 of cytoplasm,
  • in a spherical cell (10 μm radius),
  • in a cylindrical cell (1 μm radius, 100 μm long).

    • (2) The ionic concentrations inside and outside of a neuron are given below:
    Inside (mM) Outside (mM)
    K+ 168 6
    Na+ 50 337
    Cl- 41 340

  • In the presence of K+ and Na+ channel blockers, the neuron is only permeable to Cl-. If Cl- movement follows the constant field, GHK, model of current flow, sketch or compute the IV relationship for this neuron. Label all axes and mark the intersection point (for I=0 with the voltage axis) with appropriate values and units.
  • Is this inward or outward rectification?
  • What causes the rectification?
  • In the absence of K+ and Na+ blockers the permeability ratio is PK:PNa:PCl = 1:0.019:0.381. What is the resting membrane potential under these conditions?

    • (3) Figure 2 (below) shows intracellular recordings, measuring membrane potential (VM). Bath solution contained (in mM): 127 NaCl, 25 NaHCO3, 3 KCl, 1 MgSO4, 12 glucose, 1.2 NaH2PO4, and 2 CaCl2. In order to block synaptic transmission the CaCl2 was reduced to 0 mM and replaced with equimolar MnCl2. Figure 2 shows the effect of [K+]o (1 to 30 mM) on baseline membrane potential (VM). Black squares are in the presence of normal bath solution (above), white squares after replacing NaCl and KCl in the bath solution with Na-isethionate and K-isethionate, where isethionate is a large impermeant anion that substitutes for Cl-.

    The data below give (VM) as a function of [K+]o in the presence of normal versus very low (i.e., 4 mM) Cl-, and are taken from Figure 2. Answer the questions that follow:

    [K+]o VM (mV) VM (low [Cl-]o)
    1 -72 -65
    3 -71 -64
    10 -62 -54
    15 -52
    20 -44 -42.5
    25 -39
    30 -34 -28
  • Does Vm depend predominantly on K+ as predicted by the Nernst equation? Give evidence and summarize your reasoning. Assume in the experiments that [K+] was in the range of 0.4 - 50 mM.
  • What two aspects of the data in Figure 2 suggest that ions other than K+ may influence VM? Describe the biophysical mechanisms by which these ions influence VM in the context of the data shown.
  • What salt provides the 4 mM Cl- present in the bath solution after the NaCl and KCl substitution?
  • Use any computer program of your choice to plot the data (above) and illustrate your answers to the above two bulleted questions using appropriate equations. Define relative permeabilities as follows: PNa/PK = α and PCl/PK = β. You do not have to fit the equations to the data in rigorous fashion (although you certainly may do that if you wish (I did), but just making reasonable estimates of parameters α and β will be useful to demonstrate GHK theory and make your points. Assume [Na+]i = 10 mM, [Cl-]i = 6.6 mM, and [K+]i = 106 mM.

    • (4) Write you own simple RC circuit model in XPP where you deliver a step command of current at t=5 and then terminate the step command at t=30. Run your simulation from t=0 to t=50 for several levels of applied current and plot your results. Recall how to use the HEAVYSIDE step function in XPP.

    Lecture 7 Top - Cellular Biophysics and Modeling - Del Negro Homepage - Applied Science - The College of William and Mary