Model Theory

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A central goal of single channel kinetic analysis is to provide a description of the kinetic behavior of the ion channel, including estimation of transition rate constants. Single-channel kinetics are simulated by constructing a Markov model. Visual interaction with models is described in Model and Model Properties; this page is more theoretical.

Contents

States and Classes

The model describes the conformational states in which the channel can exist. Each state belongs to a class, which is defined by its amplitude, such as closed or open. For example, if a certain channel has a closed state, an open state, and another closed state (e.g., inactivated), then the channel consists of two classes - closed and open, and three states- closed1, open2, and closed3. Each colored box represents a state, and the color of the box indicates its class. There can be multiple states with the same class. The program automatically numbers states in the order in which they are added to the model. The color (class) of a state may be altered by either double-clicking the left mouse button while positioned over the state, or by right-clicking on a state, clicking Properties, and selecting a color from the State and class properties window.

A state's entry probability (P0) is the chance a record begins with that state. Entry probabilities are normalized so they sum to 1.0. The default (all states P0 = 0.0) makes all states equally likely.

Rates

If a direct transition is possible from one state to another, you connect them and define the forward and backward rate constants. A rate constant has units of per second. It's the inverse of the mean time it takes to make the transition, or the average number of transitions per second spent in the "from" state.

The formal expression representing the transition rate between states i and j (kij) is k_{ij} = k^0_{ij}Pe^{k^1_{ij}Q}, where k0 and k1 represent the parameters that are intrinsic to the model, and P (typically concentration) and Q (typically voltage) are experimental conditions -- stimuli -- specific to a data file. By defining a rate constant in terms of k0 and k1, we can fit a model to data obtained at several values of P or Q, or model the response to a changing stimulus.

When you mark a rate as P- or Q-dependent, you name the experimental variable it depends on. The default is "Ligand" or "Voltage", but you can be more descriptive. The value of "Ligand," for example, is defined in each Data file, in Data Properties under Experimental Conditions. You can also enter experimental conditions in many Action Properties windows, such as MIL Properties.

The rate constant, k0, is defined as the transition rate at unity concentration and no other stimulus. Ligand concentration can have whatever units you like, e.g. Molar or micro-Molar. k0 has the inverse units, per-second, so if Concentration is given in μM, k0 is given in \frac{1}{\mu M \cdot sec}.

The k1 term is more complicated. For voltage, it can be defined as:

k_1 = \frac{z \cdot e}{k_BT}

where z is the valence of a gating charge, e is the charge of an electron, kB is Boltzman's constant, and T is the absolute temperature (kBT / e is ~ 24 mV at 20°C). Enter Q stimuli in any appropriate unit U, and k1 will have units of \frac{1}{U}. Rates which do not respond to Q should have k1=0.

See also: How to set Q (Voltage)

It's possible to use rate constants with other units, by tricking QuB. You have to scale your data sampling rate by the same factor. For example, to interpret rate constants in per millisecond, change the Sampling in Data Properties so it reads in mHz instead of kHz -- for a file sampled at 10kHz, you'd enter Sampling as 0.01.

Constraints

QuB can maintain linear relations between rate constants while solving. It can fix a particular k0 or k1, fix the ratio between a two of them, and keep loops in detailed balance (the product of the clockwise rate constants is equal to the product of the counterclockwise rate constants). Each constraint reduces the number of free parameters by one, so the fit is faster and more reliable. You can also "enforce constraints," so that editing one rate changes all constrained rates.

See also: How to satisfy cycle (im)balance

The IdlBase algorithm also honors constraints on amplitude: 

 exc Amp is the excess amplitude, of a given class vs the black.
 In other words, the difference.
 Fix Amp constraints the absolute amp value of a given class.
 Fix exc Amp constraints the difference (excess) in amp between a given class and the black.
 Fix diff exc Amp constraints the difference in excess amp between
 any two classes. Since the excess is defined relative to the same
 class (black), it is identical to the absolute difference.
 Scale ... has the usual meaning.

Note that the amplitude constraints are implemented only in the IdlBase algorithm, for singles.

File Formats

  • MDL is QuB's old model format.
  • QMF is QuB's new model format, with support for
    • Ligand (P) and Voltage (Q) variable names
    • Constraints on Amplitude
    • Kinetic constraints with a value
    • Display properties

Tutorials

Tutorial:Basic Model

Not written

  • The A matrix
  • Peq
  • Energy landscapes


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