These models are for illustrative purposes only: they show qualitative features of kinetic scheme models but the numbers do not correspond to any real channel types. For a wider range of models, try the most recent Catacomb distribution from _EXTLINK(www.compneuro.org/catacomb, compneuro.org) or if you have Catacomb installed, use its upgrade option.
These models are built with the KSChannel component of the KSCell object. The KSChannelEditor interface lets you compose the state diagram, set the conductance of the various states and the parameters of transitions between them. The state diagram does not have to be connected. If it consists of two or more disjoint parts, these are treated as serial independent gating complexes. To include multiple copies of a complex on the channel, set the Nserial field for any one of its states to the number of copies.
There are two ways of exploring the behavior of kinetic scheme channels: with the KSStateViewEditor interface and the KSCalc object. The first does a stochastic simulation of a bunch of channels where the user controls the membrane potential. The second requires rather more background: the permeant ions must be defined, the i/v law and the intracellular and extracellular solution. KSCalc will perform voltage or current clamp recordings depending on the units set on the command waveform.
Voltage clamp recording of the channel in the previous example. Opening both examples at the same time allows the channel parameters to be varied with the voltage clamp behavior continually recalculated.
The parameters of the calculation are shown at the top. The
pre-command is applied to the channel before recording and
between steps. The v/i command is the one for which the current is
recorded. Either can be edited by checking the floating-frames
checkbox at the top and using the edit option. As with all the
windows help is available with the little ? button.
A kinetic scheme representation of a channel with Hodgkin-Huxley kinetics. There are two gating complexes corresponding to the activation and inactivation gates in the HH formalism. The relative conductance of the channel is the product of the relative conductances of the independent complexes.
The upper complex changes from a closed to an open state with depolarization. The Nserial variable indicates that there are actually three identical copies of this complex in series on the channel, corresponding to the integer power of the activation gate in the HH formalism. The lower complex is for the inactivation, and moves from an open to a closed state with depolarization.
Clicking on one of the transitions brings its parameters into the lower windows. The one on the left shows the equilbrium position between the two states (0 means all in the left hand state, 1 all in the right) as a function of the membrane potential. The middle one shows the time constant of the transition as a function of membrane potential. The parametrers of the transition are shown on the right. They may be manipulated either with the sliders or by dragging points on the graphs. After modifying the graphs you must move the mouse over the slider to see the new numerical values.
The parameters of the transition shown in this example are a superset of those
commonly used for HH models. The first two, r-f and r-r are the
forward and reverse rates (transitions per millisecond) when there is no
potential difference across the membrane. The equivalent gating charge z
sets the voltage dependence of the transition, as the charge transfer
across the membrane corresponding to the change from one state to the
other. The time constant reflects an activation barrier to the transition
between the states. The first two parameters set its height, and gamma
sets its position as a fraction of the membrane field traversed by the
putative gating particle. Extreme values of gamma (0 or 1) mean that
the transition in one direction is independent of the membrane potential:
the barrier is right at the end so tipping up the potential does not
change its height relative to the gating particle.
Finally, r-m is the saturation rate of the transition. For the mid-range
of potentials this has little effect, but at the extremes it stops the transition
rate ever exceeding r-m. This may be useful, for example where
each state is intended to represent a group of physical states of which
only one is accessible to the transition. Then whatever the potential the
effective rate would be limited by the internal rate of supply to the
Stochastic simulation of a kinetic scheme. The initial scheme is very simple, but you can add more states by tearing them off the icon at the top. Likewise, transitions can be torn off and stuck to the states. To detach them, use the right mouse button.
The display on the right shows a simulation of nchan channels at a membrane potential V mV. To start it running, move the V slider a little. You can continue to vary V as the simulation is running. The history of the potential variation is shown in black at the top, and that of the conductance change in blue. If they are off the axis, either position the mouse over the window and press the f key (for _TTfind)), or stretch the vertical scrollbar so it reaches the top of the window.
The calculation proceeds in steps of timestep ms, as far as possible, running rateFactor ms of simulation time per second of real time. The frameRate is the number of times per second the screen is redrawn. It may need adjusting to get a smooth display: too low and it is jerky, too high and it slows the calculation down.
If you make a channel with multiple gating complexes (by setting Nserial
mor than 1 or adding an isolated complex in the diagram) then the bar for each
cell is subdivided into horizontal slices for the different complexes.
Hodgkin-Huxley model as a single kinetic scheme.
Although the HH model assumes serial independent gating complexes, it is possible to represent it as a single kinetic scheme. One must enumerate all different combinations of the positions of the gates and associate a single state to each combination.
The transitions between the resulting states are no longer independent, however but are multiples of one another by various factors. This is effected in the model by making most of the transitions dependent on other transitions in the diagram for their parameters. The top row of states represent the non-inactivated states, and the bottom row the inactivated ones. The three vertical grey transitions just take their parameters from the one on the left. Click on one of the transitions: in the bottom panel it shows that the source is the first defined transition and the forward and reverse rates are copied directly (multiplied by 1).
For the horizontal transitions the situation is more complicated, as there are combinatorial terms in the factors giving values of 0.5 and 1.5 for x-fwd and x-rev.
For a channel model which fits the HH formalism exactly, implementation with
independent gating complexes is much simpler and quicker to compute.
If, however, there is some level of cooperativity for example between the
transitions so they are no longer independent this type of expanded state
diagram may be necessary.
A simple non-inactivating potassium channel.
The model contains two channels. The one displayed can be selected with the menu at the top left. The first model has four identical copies of a two state gating complex. The second model expands this to a single complex of five states, in which the rates of successive transitions depend on that of the first one.
State transitions in the potassium channel example. The channel to run
can be selected from the menu at the bottom. Other options are as for the
earlier stochastic model. The red state is the open one, and the brown the
one most favoured by deopolarization.
Voltage clamp recordings of the potassium channel models. The simulation is run for a combination of the two channels in varying densities as given under the channel density menu. You can remove one or the other by setting the density to zero (put the mouse over the slider and type 0 followed by return).
Alternatively, to get a button for setting the density to zero, put the mouse
over the slider, press the spacebar, click the the z box and then