Supplementary material for Yi et al. (2000) Proc. Natl. Acad. Sci. USA 97 (9), 4649-4653.

Barkai-Leibler Model and Derivation of Integral Control

The focal point of this model is the receptor complex, denoted E, which consists of MCP (receptor) + CheA + CheW. This complex possesses two states: active and inactive. When active, E phosphorylates the response regulator protein CheY to form CheY-P. CheY-P interacts with the flagellar motor to modulate the rotational behavior of the flagella. System activity refers to the concentration of active receptor complexes.

The receptor complex can bind ligand and is modified by methylation on m sites (m = 0, 1, . . . , M). The ligand occupied and unoccupied forms are denoted $E^{o}_{m} ,$and $E^{u}_{m} ,$, respectively. CheR methylates the receptor complex, and CheB demethylates the receptor complex. The probability that a given species of E is active depends on the methylation level and ligand occupancy of the receptor and is denoted $alpha^{i}_{m}$( $ i in {o, u}$).

With M = 2, the model is comprised of 14 differential equations representing the dynamics of the 14 receptor species in the model. Because the total concentration of receptor complex is a constant, there are only 13 independent equations. The equations are derived by the application of the principle of mass action to the model reactions depicted in Box 1 of Barkai and Leibler [Barkai, N. & Leibler, S. (1997) Nature (London) 387, 913-917]. Below, we list 3 of the 14 equations for illustrative purposes:

begin{displaymath}frac{dE_0^u}{dt} = -k_l l E_0^u +k_{-l}E_0^o - a_r alpha_0... ...a_0^u)E_0^u R + d_r {E_0^uR} + k_{b1}^u {E_1^uB} nonumber end{displaymath}

  

begin{displaymath}frac{d{E_0^uR}}{dt} = - k_l l {E_0^uR} + k_{-l} {E_0^o... ...pha_0^u)E_0^uR - d_r {E_0^uR} - k_{r0}^u{E_0^uR} nonumberend{displaymath}

  

begin{displaymath}frac{d{E_2^uB}}{dt} = -k_l l {E_2^uB} +k_{-l} {E_2^oB... ...2^u) E_2^u B - d_b {E_2^uB} - k_{b2}^u {E_2^uB}. nonumberend{displaymath}

  

${E_0^uR}$and ${E_2^uB}$represent the concentrations of the complexes between E0u and CheR, and between E2u and CheB, respectively.

We have made several modifications to the Barkai-Leibler formalism. First, we distinguish the different methylation, $k_{rm}^i ,$, and demethylation, $k_{bm}^i ,$, catalytic rate constants for the various receptor complex forms. We also include a term for the association of CheB with inactive receptor complex. In the Barkai-Leibler model, this association rate constant, $a'_b ,$, is set to 0.

To derive the integral control equation, we consider the case of two methylation sites, M = 2. Extending the result for M > 2 is straightforward. We can rearrange the equations of the Barkai-Leibler model to obtain the following:

begin{displaymath}left(frac{dE^{u}_{0}}{dt} + frac{d{E^{u}_{0}R}}{dt}rig... ...{2}}{dt} + frac{d{E^{o}_{2}B}}{dt}right)right] = nonumberend{displaymath}

  

begin{displaymath}-k_{r}({E^{u}_{0}R} + {E^{o}_{0}R} + {E^{u}_{1}R} + {... ..._{1}} + bar{E^{o}_{1}} + bar{E^{u}_{2}} + bar{E^{o}_{2}}), end{displaymath}

[3]

where $k_{r0}/k_{r1} = k_{b1}/k_{b2} = epsilon$, $k_r = k_{r0} ,$, and $k_b = k_{b1} ,$. For this simplification, we exploit the fact that CheB demethylates only active receptor complexes (assumption 1) and that CheB operates under Michaelis-Menten kinetics where Kb, the Michaelis constant of CheB, is independent of ligand occupancy and the methylation state (assumption 2). As a result, we can substitute the expression $B bar{E}_m^i/K_{b}$for the term $ {E_m^iB} $, where $bar{E}$represents the active receptor complexes. Assumption 2 helps to define $epsilon$( $epsilon = k_{b1}gamma/k_{r1}, gamma = k_r/k_b$) and allows one to pull out the factors kr and kb from the right side of Eq. 3.

We can define the total activity of the receptor complex to be ( A = bar{E^{u}_{1}} + bar{E^{o}_{1}} + bar{E^{u}_{2}} + bar{E^{o}_{2}} ), because E0 is inactive (assumption 3). We now set

begin{displaymath}dot{z} = mbox{boldmath$kdot{x}$} = left(frac{dE^{u}_{... ..._{2}}{dt} + frac{d{E^{o}_{2}B}}{dt}right)right], nonumberend{displaymath}

  

and let

begin{displaymath}y = -k_{r}({E^{u}_{0}R} + {E^{o}_{0}R} + {E^{u}_{1}R... ...ar{E^{o}_{1}} + bar{E^{u}_{2}} + bar{E^{o}_{2}}). nonumberend{displaymath}

  


Then we have the following result, where Rbnd denotes bound CheR:

begin{displaymath}dot{z} = -k_{r}R_{bnd} + frac{k_{b}}{K_{b}}B A = frac{k_{b}}{K_{b}}B (A - frac{k_{r}R_{bnd}K_{b}}{k_{b}B}) = y. end{displaymath}

[4]

This integral control equation indicates that at equilibrium, the activity of the system will tend toward a fixed steady-state value, $ A^{st} = k_{r}R_{bnd}K_{b}/k_{b} B ,$. Finally, noting that $B_{bnd} = k_r R_{bnd}/k_b ,$, we can substitute $(B_{tot} - k_r R_{bnd}/k_b) ,$for B in the above equation and also replace kr/kb with $gamma$, resulting in the expression for Ast given in Eq. 2. Ast is independent of the ligand concentration if Rbnd is independent of ligand concentration (assumption 4), and thus perfect adaptation holds.

We can obtain a more intuitive feeling for z by setting $epsilon = 1$, and then $dot{z} = d(E_0^{tot} - E_2^{tot})/dt$, where $E_n^{tot} ,$denotes all receptor species possessing n methyl groups. Because $(E_0^{tot} + E_1^{tot} + E_2^{tot}) ,$equals the total number of receptors, $-dot{z} = d(E_1^{tot} + 2E_2^{tot})/dt = d(sum_{k=0}^{2} kE_k^{tot})/dt$. Thus, -z roughly represents the total methylation level of the receptors.