Page 36 - YORAM RUDY BOOK FINAL

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The computed electrostatic energies at various S4 positions during its gating movement

generate an energy landscape (energy as a function of translation and rotation of S4). The
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KCNQ1 energy landscape is shown in Figure 2.20F, bottom; it contains three energy minima
(deep blue) that correspond to stable conformations. These minima are labeled (right to left)
deep closed state, intermediate closed state, and a permissive state of the voltage sensor (note

correspondence with the two-stage voltage sensor activation described in conjunction with
Figure 2.9). Protein conformations associated with these states are shown above the energy
landscape (Movie 1 shows the motion of S4 here: https://youtu.be/vamDvoj-BuQ). Note that
channels can open only when all four voltage sensors are in the permissive state (concerted or

cooperative gating).

The positive charges on S4 and their movement across the membrane make the channel
gating V dependent. The change in the energy landscape due to V is computed using a
m
m
modified Poisson-Boltzmann equation (Poisson-Boltzmann-Voltage equation, or PB-V) . In the
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presence of negative V = -80mV, the closed states are stabilized (Figure 2.21C). With application
m
of positive V (+60mV) the stable region (minima) shifts toward the permissive state and the open
m
state is favored.

The probability of movement between conformations can be computed using the
Smoluchowski equation: 107,111,112

∂ Ρ ( ) tx, = ∂ D  ∂ Ρ ( )− Ftx, β ( ) ( ) (2.8)

Ρ x,
t
x

∂t ∂x  ∂x 
Where P(x,t) is the probability of occupying position X at time t, D is the diffusion constant,
F(x) is the force acting on the moving particle (introduced by the energy landscape) and ß = 1/kT

(k is the Boltzmann constant). From this equation, transition rates between neighboring states
can be obtained. Figure 2.21D shows a Markov model of KCNQ1 in which the energy landscapes of
the four subunits determine residency in the permissive state. Employing a Monte Carlo
simulation, each voltage sensor performs a random walk on its own energy landscape. Each

subunit is initialized to a position on the energy landscape. During a given time step, dt, the
voltage sensor can make a transition from its current position to an adjacent position only if a
random number (between 0 and 1) is less than the product of dt and the transition rate
(computed using the Smoluchowski equation). Each time all four voltage sensors are in the

permissive state (to the left of the white dashed line on the energy landscape) the channel can
make a cooperative transition to the open state O (Figure 2.21D). Once in O , channel transitions
1 1
between the open states (O -O ) and between O and the inactivation state (I) are simulated with
1 5 5
a Markov model. The presence of five open states is inferred from a delay before the onset of

inactivation and an increase in the time constant of deactivation that is proportional to the
duration of a depolarizing pulse. A flickery blocked state is included in the model based on
experimental evidence from noise variance analysis and studies in the presence of rubidium .
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