Binding

Free energy of a solution

Free energy of a solution The free energy of a “solution” depends on its concentration It becomes progressively harder and harder (i.e. energetically costly) to force more solute into a limited space This is why solutes don’t spontaneously aggregate in one location We will eventually apply this to the “reaction” A + B ⇆ C, which is shorthand for the actual reaction nA A + nB B + nC C ⇆ (nA-1) A + (nB-1) B + (nC+1) C There is an entropic component to solvation (PBoC section 6.2): Imagine a volume of solvent as comprising N “boxes” in which to put n particles of solute (N = V/h3 if necessary). There are W = N!/ [n! (N-n)!] ways to do this, so the Boltzmann entropy S = k ln W is

S = kB ln

!

N! n!(N − n)!

"

In general this expression is complicated, but for n ε + εR

This is just like the hypothetical phosphorylation energy landscape.

To make it cooperative, assume that either all subunits switch from T to R or none switches (“symmetric” model) This is interesting because the actual O2 sites are far from each other. Binding in one subunit changes the energy of binding in another subunit, far away. This is called allostery.

PBoC illustrates this for a 2-subunit “Hb”: Without the all-or-none requirement, there would be 16 possible states instead of 8.

Multistate systems The MWC energy is E(σm , σ1 , σ2 , σ3 , σ4 ) = (1 − σm )"T σm=0 for the T state and σm=1 for the R state

4 !

σi + σm

i=1

"

" + "R

4 ! i=1

σi

#

There is only one σm for all four subunits (unlike the σi’s) because the entire tetramer is in the T or R state. This notation is what makes the system cooperative. Very subtle. If we had four σm’s (σm,1 .. σm,4) this would not be a cooperative system. That is, there would be only one Kd regardless of how many O2 were bound.

Within each state T or R, oxygens bind completely independently, so the partition function is

Z = (1 + x)4 + e−β" (1 + y)4

x = e−β("T −µ) =

c −β("T −µ0 ) e c0

y = e−β("R −µ) =

c −β("R −µ0 ) e c0

This form of the partition function Z is sometimes called the binding polynomial. N = kBT ∂(ln Z)/∂µ. After some algebra,

best fit of Imai data to MWC model

4

(1 + x)3 x + e−β" (1 + y)3 y !N " = 4 (1 + x)4 + e−β" (1 + y)4

3

2.5 number bound

The best fit (green) to binding data (blue) is much better than a non-cooperative model (red).

3.5

2

1.5

1

0.5

0 −1 10

0

10

1

10 ppO2

2

10

3

10

Multistate systems Pauling model (PBoC p. 274) The sites are not independent, so the binding energy depends on whether adjacent sites are already bound (“sequential model”) The model assumes that Hb is a pyramid, so any two sites are adjacent. (Which is not obvious from the PBoC illustration).

The Pauling energy is

4 !

4 J !! E(σ1 , σ2 , σ3 , σ4 ) = " σi + σi σj 2 i=1 i=1 j!=i

The cross term (J term) contains the cooperative interaction.

This is a finite-size version of the most famous system in statistical mechanics: the Ising model.

From this energy, the partition function is (PBoC 7.44)

Z = 1 + 4e−β("−µ) + 6e−2β("−µ)−βJ + 4e−3β("−µ)−3βJ + e−4β("−µ)−6βJ and the mean occupancy is N = kBT ∂(ln Z)/∂µ:

4x + 12x2 j + 12x3 j 3 + 4x4 j 6 !N " = 1 + 4x + 6x2 j + 4x3 j 3 + x4 j 6

−βJ x ≡ e−β("−µ) = ce−β("−µ0 ) j ≡ e

Actually, there’s a more involved Adair model that includes 3-body interactions (energy K) and 4-body interactions (energy L). See PBoC p. 277-278 for details). This doesn’t fit the oxygen binding data any better, but can account for competitive binding with, for example, CO.

Multistate systems The end result (with physiological parameters) is a suppression of the intermediate states that have 2 or 3 O2 bound:

Qualitatively, you can bind one O2, but as soon as you bind the second the third and fourth immediately bind as well, giving a Hill coefficient of about 3. As you suppress intermediate states you approach the all-or-none behavior of the Hill function (as soon as one ligand binds, they all bind) You can interpret this as Kd’s that decrease with the number of oxygen bound, so binding the first O2 makes it easier to bind the next one.

ΔE1

ΔE1

ΔE1

Multistate systems All these models can be parameterized either by energies or by Keq (or, equivalently, Kd). Chemists tend to use Keq, which are more easily measured. Physicists tend to use energy, which make the relationship between different states more transparent. The various Keq are not independent; this seems more obvious to me in the energy picture.

ΔE1

ΔE3

ΔE2

ΔE4

[TAB ] = KT A = e−β(ET +A −ET ) = e−β(∆E1 ) [TB ][A] What is the equilibrium for the diagonal? From the energy picture, it is

[RAB ] = e−β(∆E1 +∆E2 ) = e−β(∆E3 +∆E4 ) [TB ][A]

From the binding equilibrium picture it is

[RAB ] = KT A YAB = YB KRA [TB ][A]

Multistate systems If the Keq are interpreted as microscopic equilibrium constants, then we need to account for the different number of available binding sites in each intermediate state For n identical binding sites with identical microscopic binding equilibria KT and KR for the T and R states, the macroscopic equilibiums constant for 1-binding, 2-binding, etc. are all different:

Multistate systems Flagellar rotary motor The motor of the bacterial flagellum drives it full speed CW or CCW. This looks very much like a two-state system.

The probability of going CW or CCW is modulated by the binding of a signaling protein. There are many (perhaps 34) binding sites for CheY on the motor

The motor is like a 34-subunit Hb whose allosteric state you can observe from outside Response regulatorthe output in bacterial cell.

Hill function with n=2.5, Kd=18.8 µm

From Alon et al, “Response regulator output in bacterial chemotaxis”, EMBO J 17:4238

Fig. 1. Fraction of time tethered cells spend turning CW (fCW). Cells induced with various amounts of IPTG were tethered and their motion was analyzed using video tracking. Each point is an average of data

Fig. 3. Distribution of CW bias between different individ Plotted is the fraction of time spent CW, fCW, divided by fCW over all cells with the same CheY induction level, "

Multistate systems Flagellar motor Make a motor model analogous to the MWC model U.Alon et al. re-constitute a full bundle. A crude estimate of the typical time for such hydrodynamic interactions is the bacterium N length (a few micrometers) N divided by its velocity (10– 0 20 µm/s), yielding a tumble duration of the order of tenths of seconds. N One may view the flagella interactions as an additional level of information processing in the chemotaxis response. The result of this processing is that whereas CW N individual motors displayN prolonged CW intervals, swim0 ming cells keep their tumble duration short and independent of P-CheY. This behavior of swimming cells is close to the expected optimal behavior. Since it is only during runs that the bacteria sample their environment and reach 7 Assuming that L0=10 and N=30 gives K=7.6 2.0should µM beand favorable regions, each± tumble kept as brief as is necessary to randomize the heading of the subsequent run. Ψ=2.0 ± 0.2. An additional difference between tethered motors and free-swimming cells is observed at very high P-CheY The CheY-P affinity betweenlevels. CW Whereas and CCW differs by only ΔG CW rotation increases monotonically (Figure 1), swimming cells become less tumbly at P-CheY = kBT ln Ψ = 1.4 kBT: small. levels higher than ~50 µM (Figures 4 and 5). This is probably caused by the runs associated with CW rotating The increased affinity has to be because we needin aflagella bundles thatsmall have previously been documented switch mutants that lock the motor in an extreme CW Hill coefficient of 2.5 though we have 30 binding sites. state (Khan et al., 1978; Wolfe and Berg, 1989). The average run speed drops abruptly at high P-CheY levels to ~50% of the wild-type run speed (Figure 6), in agreement with the observation that speed generated by CW bundles is about half the speed generated by CCW bundles (Khan et al., 1978). It should be noted that CW runs only predominate at P-CheY levels that exceed the total CheY content of wild-type cells. At all levels of P-CheY that could possibly be attained in wild-type cells (i.e. 0 to ~50 µM), the tumbling frequency increases with increasing P-CheY. It is only at P-CheY levels exceeding the maximal physiological level that the tumbling frequency begins to decrease with increasing P-CheY. This latter behavior can be detrimental since it leads to reverse chemotaxis where cells move towards repellants and away from attractants (Khan et al., 1978). Strikingly, the cell appears to have tuned the signal transduction pathway (kinase activity and CheY concentration) to give the highest possible maximal tumbling response that does not carry over to the reverse chemotaxis regime.

Z = L (1 + c/K) + (1 + Ψc/K) =

(1 + Ψc/K) L (1 + c/K) + (1 + Ψc/K)

microscopic Kd’s

p

Fig. 9. Allosteric model for the motor switch. The motor is assumed to be in one of two states, CCW or CW. It displays spontaneous stochastic transitions between the states, with a (large) equilibrium constant L0. The motor is assumed to have N independent, identical binding sites (represented here by open circles) for P-CheY (stars). Both of the rotation states can display multiple P-CheY occupancy states, with between 0 and N sites bound by P-CheY proteins. The probability of P-CheY binding is not the same for the two states. The equilibrium dissociation constants for P-CheY binding are K for binding to the CCW state and K/! for binding to the CW state (vertical arrows). Transitions between CW and CCW states with m binding sites occupied by P-CheY occur with equilibrium constants Lm (horizontal arrows). In the model, P-CheY binds preferentially to the CW state (! "1), and thus increasing cytoplasmic concentrations of P-CheY shift the equilibrium towards a higher fraction of time spent

Multistate systems RNA hairpins Some systems that look like they should be multistate are so cooperative that they are effectively two-state. Pull on a RNA hairpin with an optical trap, measuring the extension as a function of force.

Multistate systems RNA hairpins An experimental trace looks like

Multistate systems RNA hairpins This fits beautifully to a two-state model of RNA unzipping. Note that there is only one free parameter in this model (ΔF0). Once you pick ΔF0 to give the proper midpoint of the folding curve, the rest of the curve is mandated Because, just like we saw for simple ligand binding, there is only one family of curves for two-state binding. Once the pick the Keq (here it’s called Feq = ΔF0), there is no freedom to change the shape of the curve any further.