In Sect. 3.1 we described the classical Monod-Wyman-Changeux (MWC) model of cooperative receptor-ligand binding, and showed how cooperativity can sharpen the stimulus-response properties of receptors to ligand concentration. (For a general review of cooperativity in cell signaling see Ref. [2].) In particular, the sharpness of the sigmoidal response is related to the number of interacting sites. Cooperative binding was also hypothesized to be one mechanism for signal amplification in bacterial chemotaxis, see Sect. 5.3. A characteristic feature of amplification via cooperative binding is that it occurs at thermodynamic equilibrium.
In this section we consider an alternative, non-equilibrium mechanism for the generation of sigmoidal response curves, based on a driven phosphorylation dephosphorylation cycle (PdPC) [13, 2]. The system consists of a protein that can exist in an unmodified (unphosphorylated) form known as the substrate and a modified (phosphorylated) form known as the product. Interconversion of the inactivated and activated protein states is catalyzed by two enzymes, kinases that phosphorylate the inactivated protein and phosphotases that dephopshorylate the activated protein.
PdPC is a very common signaling mechanism within cells, and also occurs within the context of intracellular protein gradients (see Sect. 9.1). The analysis of PdPCs proceeds along analogous lines to the derivation of Michaelis-Menten kinetics in a single enzymatic pathway, see Box 6B. Ultrasensitivity refers to the fact that the
PdPC can exhibit an output response that is more sensitive to changes in stimulus than the Michaelis-Menten form.
The cooperative nature of the underlying non-equilibrium steady (or stationary) state (NESS) has been explored in considerable detail by Qian and collaborators
[27, 11, 11, 12, 1, 7, 12]. In particular, they highlight how the PdPC can be considered as a form of temporal cooperativity. Substrate proteins compete for kinase such that, as the reaction proceeds, the substrate number decreases and the competition lessens. Thus, earlier turnovers (phosphorylation) help the later turnovers so that the substrates are temporally cooperative. However, one also has to take account of the competition from the product (phosphorylated proteins) for the same enzyme.
This is precisely the role of the driven system: If the products were equally likely to compete for the enzymes, i.e., the enzymatic reactions were fully reversible, then the temporal cooperativity would disappear. The system is maintained out of equilibrium by ATP. (In the original analysis of PdPCs, the final step in the enzymatic reaction schemes was taken to be irreversible [13].)
Ultrasensitivity in a PdPC was originally analyzed by Golbeter and Koshland [13], who assumed that the action of kinases and phosphotases are irreversible. Suppose
1

that a protein exists in the unmodified form W and the modified form W
∗
. Interconversion of the two form is catalyzed by two enzymes E
1 reaction schemes and E
2 according to the
W + E
1
W
∗
+ E
2 a
1 d
1 a
2
W E
1
→ W
∗
+ E
1
, d
2
W
∗
E
2 k
→ W + E
2
.
(5C.1a)
(5C.1b)
[
Introducing the concentrations w = [ W ] , w
∗
W
= [ W
∗
E
2
] , the corresponding kinetic equations are
∗
] , z j
= [ E j
] , w
1
= [ W E
1
] and w
∗
2
= dw
= − a
1 wz
1
+ d
1 w
1
+ k
2 w
∗
2 dt dw
1
= a
1 wz
1
− ( d
1
+ k
1
) w
1 dt dw
∗
= − a
2 w
∗ z
2
+ d
2 w
∗
2
+ k
1 w
1 dt dw
∗
2 dt
= a
2 w
∗ z
2
− ( d
2
+ k
2
) w
∗
2
.
These equations are supplemented by the conservation equations
W
T
= w + w
∗
+ w
1
+ w
∗
2
,
E
1 T
= z
1
+ w
1
,
E
2 T
= z
2
+ w
∗
2
.
(5C.2a)
(5C.2b)
(5C.2c)
(5C.2d)
(5C.3a)
(5C.3b)
(5C.3c)
Proceeding along analogous lines to the derivation of Michaelis-Menten kinetics
(see Box 6B), we assume that the concentration of W and W
∗ is much larger than that of the kinase and phoshphotase, that is, W
T w + w
∗
E
1 T
+ E
2 T or equivalently W
T
=
. This implies that the time scale for the dynamics of the enzymes E
1 and E
2 is much faster than that for the dynamics of W and W
∗
. Performing a separation of time-scales, we can treat the concentrations w and w
∗ as constants when analyzing equations (5C.2b,d), while we can take the steady-state values of the concentrations z
1
, z
2 when solving equations (5C.2a,c). Hence, setting dw
1 w
1 z
2 in (5C.2b) we can solve for w
1
= E
2 T
− w
∗
2
/ dt = 0 and in terms of w . Similarly, setting dw in (5C.2d) we can solve for w
∗
2 in terms of w
∗
∗
2 z
1
= E
1 T
−
/ dt = 0 and
. We thus obtain the reduced kinetic scheme
W v
1
( w )
W
∗
, (5C.4) v
2
( w
∗ ) with v
1
( w ) =
V
1 w / K
1
1 + w / K
1
, v
2
( w
∗
) =
V
2 w
∗ / K
2
1 + w
∗ / K
2
, (5C.5) and
V
1
= k
1
E
1 T
, V
2
= k
2
E
2 T
, K
1
= d
1
+ k
1
, K
2 a
1
= d
2
+ k
2
.
a
2
2

1.0
0.8
0.6
0.4
0.2
K
1
= K
2
=10
-2
K
1
= K
2
=1
10
-2
10
-1 1
V
1
/V
2
10 10
2
Fig. 5C.1: Molar fraction of modified protein W
∗ rates.
at steady-state as a function of the modification
Imposing the conservation condition W
T kinetic equation dw
∗ dt
= v
1
( W
T
= w + w
∗
− w
∗
) − v thus yields the single independent
2
( w
∗
) .
(5C.6)
The steady-state fraction of the product protein W solving the equation w
∗ v
1
( W with positive solution
T
− w
∗
) = v
2
( w
∗
∗
, φ
= w
∗
/ W
T is thus obtained by
) , which yields a quadratic equation for
φ
=
− B +
√
B 2 + 4 AC
,
2 A
(5C.7) for V
1
= V
2
, where
A =
V
1
V
2
1
− 1 , B =
W
T
K
1
+ K
2
V
1
V
2
−
V
1
V
2
− 1 , C =
K
2
W
T
V
V
1
2
A plot of the steady-state molar fraction of W
∗
, φ , as a function of the ratio
V
1
/ V
2 is shown in Fig. 5C.1 for K
1
= K
2
. Note that V
1
/ V
2 represents the ratio of the kinase activity to phosphotase activity, which characterizes the stimulus into the
PcPC. At low values of K
1 and K
2
, there is a sharp change from low to high levels of modified protein over a very small change in the V
1
/ V
2 ratio (ultrasensitivity); this corresponds to a regime in which the two enzymes are saturated. On the other hand, for large values of K
1 and K
2
, the curve is relatively shallow, and one obtains a response similar to first-order kinetics.
3

One of the major simplifications of the Goldbetter-Koshland model is the irreversibility of the distinct phosphorylation and dephosphorylation processes. In order to develop a deeper understanding of kinetic models of PcPCs and the nature of the non-equilibrium steady-state, it is necessary to treat these processes as reversible and to include the effects of ATP [27, 11]. Explicitly incorporating the reversible hydrolysis reaction ATP ADP + P, we have
W + E
1
+ AT P a
0
1 d
1
W · E
1
· ATP k
1 q
0
1
→ W
∗
+ E
1
+ ADP ,
W
∗
+ E
2 a
2 d
2
W
∗
E
2 k
2
→ W + E
2
+ P .
q
0
2
(5C.8)
(5C.9)
At constant concentrations for ADT, ADP and P, which are assumed to be sufficiently below the saturation levels of their respective enzymes, we can eliminate the dynamics of hydrolysis, and obtain the reduced reversible reaction scheme
W + E
1
+ AT P a
1 d
1 k
1
W E
1 q
1
W
∗
+ E
1
,
W
∗
+ E
2 a
2 d
2
W
∗
E
2 k
2 q
2
W + E
2
, with effective first-order rate constants
(5C.10)
(5C.11) a
1
= a
0
1
[ ATP ] , q
1
= q
0
1
[ ADP ] , q
2
= q
0
2
[ P ] .
If the concentrations of ATP, ATD and P were allowed to reach thermodynamic equilibrium, then the law of mass action would yield the following equilibrium constant for ATP hydrolysis:
[ ATP ] eq
[ ATP ] eq
[ ATP ] eq
= d
1 q
0
1 d
2 q
0
2 a
0
1 k
1 a
0
2 k
2
= e
−
∆
G
0
/ k
B
T
, where ∆ G
0 is the standard free-energy change for the ATP hydrolysis reaction (see also Sect. 4.3). However, since these concentrations are maintained out of equilibrium, we find that summing up the changes in free energy associated with the four reversible reactions gives
∆
G = k
B
T ln
γ
≡ k
B
T ln a
1 k
1 a
2 k
2 d
1 q
1 d
2 q
2
=
∆
G
0
+ k
B
T ln
[ ATP ]
[ ATP ][ ATP ]
.
(5C.12)
Thermodynamic equilibrium is recovered when
∆
G = 0, that is,
γ
= 1
The analysis of the reversible model proceeds along similar lines to the Goldbeter-
Koshland model. We now obtain a reduced reaction scheme of the form
4

W f
1
( w , w
∗ ) f
2
( w ∗ , w )
W
∗
, (5C.13) with f
1
( w , w
∗
) =
1 + w
V
1 w / K
1
/ K
1
+ w
∗ / K
∗
1
+
V
2
∗ w / K
∗
2
1 + w / K
∗
2
+ w
∗ / K
2
, f
2
( w
∗
, w ) =
V
2 w
∗
/ K
2
1 + w / K
∗
2
+ w
∗ / K
2
+
V
∗
1 w / K
∗
1
1 + w / K
1
+ w
∗ / K
∗
1
,
(5C.14)
(5C.15) and
V
1
∗
= d
1
E
1 T
, V
2
∗
= d
2
E
2 T
, K
∗
1
=
The steady-state molar fraction of protein W
∗ quadratic equation arising from [27] d
1
+ k
1 q
1
, K
∗
2
= d
2
+ k
2
.
q
2 is then determined by solving the f
1
( W
T
− w
∗
, w
∗
) = f
2
( w
∗
, w
T
− w
∗
) .
Note the results for the Golbetter-Koshland model are recovered in the limits q
1
, q
2
→ 0, for which K
∗
1
, K
∗
2
→
∞
,
γ
→
∞
, and f
1
( w , w
∗
) → v
1
( w ) , f
2
( w
∗
, w ) → v
2
( w
∗
) .
For finite q
1
, q
2
K
1
, K
2 one still observes ultrasensitivity in the saturated regime where are sufficiently small.
Recall that cooperative receptor-ligand binding is based on a high-order reaction scheme (see Sect. 3.1),
R + nL
K n
R n
, where K is the dissociation constant for a receptor binding a single ligand, and n is the number of binding sites. The equilibrium fraction of receptors with all their sites bound is then
Y =
[ R n
]
[ R n
] + [ R ]
=
[ L ] n
K n + [ L ] n
, and the steepness of the response curve increases with n . In particular, d ln Y n = 2 d ln [ L ]
Y = 1 / 2
.
On the other hand, ultrasensitivity in a PdPC occurs in a regime where the enzymes are saturated, which means that the steady-state is only weakly dependent on the concentrations w , w
∗
. Defining an effective Hill coefficient by n
H
= 2 d ln
φ d ln σ
φ
= 1 / 2
,
σ
=
V
1
,
V
2
5

we find that n
H
1. A Hill function would be obtained from the corresponding steady-state equation k p w ν = k d w ∗ ν , where k p and k d are the rates of phosphorylation and dephosphorylation, and
ν
=
1 / n
H
≈ 0. In a standard single-step reaction the order
ν of the reaction represents the number of molecules converted (stochiometry) in a single step, so that for
ν
≈
0 we have a “zeroth-order” reaction. Another way to view this is that converting one protein W to W
∗ will require on average n
H phosphorylation cycles, that is, n binding sites in equilibrium cooperative binding has been replaced by n
H temporal cycles in a non-equilibrium process, leading to the notion of temporal cooperativity.
A number of recent studies have investigated noise signal amplification in ultrasensitive signal transduction pathways based on stochastic versions of equation (5C.6), in which the concentration of kinases or phosphotases fluctuates (intrinsic noise) or theres is a fluctuating source term (extrinsic noise) [2, 14, 11, 9]. A full analysis would need to start from a stochastic version of the full system of Eqs. (5C.2).
Here, however, we will follow along similar lines to previous authors and consider the effects of fluctuations by applying linear response theory to equation (5C.6), see
Sect. 2.2.5. For concreteness, suppose that the concentration of kinases fluctuates by writing E
1 T
→ E
1 T
+ R ( t ) with R ( t ) evolving according to the Langevin equation dR dt
= −
γ r
R ( t ) +
σ ξ
( t ) , (5C.16) with
ξ
( t ) a white noise process h
ξ
( t ) i = 0 , h
ξ
( t )
ξ
( t
0
) i =
δ
( t − t
0
) , and σ the noise intensity. Here γ r is the rate of relaxation to the equilibrium kinase concentration E
1 T
. It is convenient to rewrite equation (5C.6) as dw
∗ dt
= F ( w
∗
, R ) ≡ k
1
[ E
1 T
K
1
+
+ [
R
W
](
T
W
T
− w
∗ )
− w
∗ ]
− k
2
E
12 w
∗
.
K
2
+ w
∗
(5C.17)
Linearizing equation (5C.17) about the fixed point solution by setting w
∗ w
∗ eq
=
φ
( V ( u )) , gives dW dt
= −
β 1
W +
β 2
R
= w
∗ eq
+ W ,
(5C.18) with
β 1
= −
∂
F
∂ w
∗ eq
=
[ K
1 k
1
K
1
E
1 T
+ [ W
T
− w
∗ eq
]] 2
+ k
2
K
2
E
2 T
[ K
2
+ w
∗ eq
] 2
(5C.19)
6

and
β 2
=
∂
F
∂ R eq
=
K k
1
( W
T
1
+ [ W
− w
∗ eq
)
T
− w ]
(5C.20)
Note that the gain of the equilibrium system, which is proportional to the slope of the input-output curves in Fig. 5C.1 is g =
∆
W / w
∗ eq
∆
R / E
1 T
=
β
2
E
1 T
.
β 1 w
(5C.21)
In order to determine the variance of the concentration w
∗
, we use Fourier transforms. Taking e
(
ω
) =
Z
∞
−
∞ e i
ω t
R ( t ) dt etc., we Fourier transform the linear equations (5C.16) and (5C.18) to obtain e
(
ω
) =
β
2
β 1
+ i
ω e
(
ω
) , R (
ω
) =
σ
0 k
−
+ k
+ u + i
ω e
(
ω
) , where e
(
ω
) is the Fourier transform of a white noise process with h e
(
ω
) i = 0 , h e
(
ω
) e
(
ω
0 ) i = 2 π δ
(
ω
−
ω
0
) .
Using the Wiener-Khinchine theorem (Sect. 2.2.5), the variance of the kinase concentration is given by the integral of the power spectrum defined by
2
π S
R
(
ω
)
δ
(
ω
−
ω
0
) = h e (
ω
) R (
ω
0
) i
That is,
σ
2
R
=
Z
∞
−
∞
S
R
(
ω
) d ω
2
π
=
Z
∞
−
∞
γ r
2
σ
2
0
+
ω
2 d ω
2
π
=
σ
2
0
2
γ r
.
(5C.22)
Similarly, the variance of the product concentration is
σ
2
W
=
=
Z
∞
−
∞
S
W
(
ω
) d
ω
2 π
β
2
1
β
2
2
−
γ r
2
σ
2
γ
2
0 r
+
=
β
2
2
2
β 1
Z
∞
−
∞
β
2
2
β
2
1
+
ω
2
γ r
2
σ
2
0
−
β
2
1
=
γ r
2
σ
β
2
2
σ
2
0
2
β 1 γ r
2
0
+
ω
2
1 d
2
ω
π
γ r
+
β 1
.
(5C.23)
If we interpret
σ W
/ w
∗ eq as the relative noise intensity of the output and
σ R
/ E
1 T as the relative noise intensity of the output, then the noise amplification of the PdPC in response to receptor fluctuations is defined by [14]
G =
σ W
/ w
∗ eq
σ R
/ E
1 T
=
E
1 T w s
β
2
2
1
β 1 γ r
+
β 1
= g s
β
1
γ r
+
β 1
.
(5C.24)
7

It follows that if the relaxation rate γ r of the receptor noise fluctuations is much slower than the relaxation rate of the protein W
∗
, then the noise amplification G approaches the deterministic gain g . This implies that ultrasensitivity produces high noise amplification in a the domain around V
1
= V ation rate of the fluctuations is relatively fast, then
2
. On the other hand, if the relax-
G ≈ g s
β 1
γ r g .
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8