TI - The cooperative model . AB - Recent experiments strongly suggest that the intracellular chemotaxis network of E.coli has a more complicated topology than that of the canonical push-pull network discussed in the previous section . In particular , in the canonical model discussed above the phosphatase reactions were described by simple Michaelis-Menten reactions . However , experiments of Eisenbach and coworkers [4] , [6] and Silversmith et al .lt @@@@@ gt [7] have shown that the activity of CheZ depends in a cooperative manner on the CheYp concentration . It is clearly important to understand how the response curve is affected by the cooperative dependence of phosphatase activity on CheYp concentration . In this section , we present a simple model for the cooperative dependence of the phosphatase activity on CheYp concentration , which can be solved analytically . Furthermore , we show that incorporation of cooperativity into the phosphatase reactions can lead to a model of type I ( see Figure 1 ) and therefore gives a possible explanation for the experiments by Vaknin and Berg [2] . In vitro data [4] , [6] , [7] suggest that the activity of CheZ depends in a cooperative manner on the CheYp concentration . The experiments of Eisenbach and coworkers [4] , [6] suggest that the activity of CheZ also depends in a cooperative manner on the CheZ concentration , suggesting that CheZ may oligomerize upon CheYp binding [4] -[6] . Other biochemical in vitro experiments [16] and more recent in vivo FRET experiments [9] , however , do not provide support for this idea . We therefore assume that the activity of CheZ in the mutant cells only depends cooperatively on the CheYp concentration . The model for the cooperative dePHOSphorylation of CheYp by CheZ is based upon the following assumptions : . . . . . . 1 ) a single CheZ dimer can bind up to two CheYp molecules ; 2 ) CheZ can dephosphorylaTE CheYp in both CheYp-bound states , thus dePHOSphorylation can occur when only a single CheYp molecule is bound or when two CheYp molecules are bound . This model can be described by two coupled Michaelis-Menten reactions , those of Eq. 3 in combination with . (5) In steady state , the phosphatase activity is given by (6) where is the total concentration of CheZ and and are the Michaelis-Menten constants of Equation 3 and Equation 5 , respectively ( see Text S3 for a derivation ) . It can be seen that if and if , the dePHOSphorylation rate is given by (7) This is a Hill function with a Hill coefficient of 2 and a concentration at which the rate is half maximal ( the inflection point ) given by . Clearly , strong cooperativity arises when 3 ) the binding of the first SUBstrate molecule facilitates the binding of the second one , making . and 4 ) the catalytic activity is higher when two SUBstrate molecules are bound than when one is bound , i.e . . In Text S3 we give an extended analysis of this model , which shows that it can fit the in vitro data of Blat and Eisenbach [6] not only qualitatively , but also quantitatively ; this fit satisfies criteria 3 ) and 4 ) . Recently , Silversmith et al .independently developed a similar model as that of Eqs. 3 and 5 on the basis of their in vitro experiments [7] , although they did not present the analytical result of Eq. 6 [7] . Interestingly , their model also satisfies criterion 3 ) : . . . . . . .binding of the first CheYp molecule facilitates the binding of the second CheYp molecule . However , in their model binding of the second CheYp molecule does not enhance the catalytic activity of CheZ [7] , in contrast to our model . We cannot obtain a good fit to the in vitro data of Eisenbach and coworkers [6] , nor , as discussed below , to the in vivo data of Vaknin and Berg [2] , without relaxing criterion 4 ) . Finally , we would like to emphasize that the rate constants derived from fitting in vivo data may differ from those obtained from fitting in vitro data . In particular , diffusion-limited reaction rates will often be lower in living cells due to a lower diffusion constant , and a detailed analysis of this model ( see Text S3 ) suggests that in this system this might be the case . In the model presented in this section , we assume that in wild-type cells all CheZ proteins are localized at the receptor cluster , while in the CheZ mutant cells all CheZ proteins freely diffusive in the cytoplasm . For both cells , the chemical reactions are given by Eqs. 1-3 and Eq. 5 . However , while the rate constants of the phosphorylaTION reactions in Eqs. 1 and 2 are identical for both cells , they differ for the dePHOSphorylation reactions of Eqs. 3 and 5 . In particular , in order to obtain a good fit to the FRET data [2] , we have to assume that in the CheZ mutant cells CheZ acts cooperatively , while in the wild-type cells CheZ acts non-cooperatively . Specifically , while for the wild-type cells , not only the two CheYp-CheZ association rates and , but also the two catalytic activities and can be assumed to be identical - - ; - - , for the CheZ mutant cells it is required that and ( see caption of Figure 3 for parameter values ) . The results for this model are shown in Figure 3 . The FRET response of wild-type cells is similar to that in the canonical model discussed in the previous section ; it is essentially linear in over the relevant range of , because CheZ acts non-cooperatively . However , the FRET response of wild-type cells is weaker than that of CheZ mutant cells over this range . This is because the catalytic activity of CheZ with one CheYp molecule bound , , is higher in wild-type cells than in CheZ mutant cells . Indeed , this model would suggest that the interaction of CheA with CheZ enhances the catalytic activity of CheZ when one CheYp molecule is bound to CheZ . Another important point to note is that the FRET response of CheZ mutant cells is strongly concave over the relevant range of . This model is indeed an example of type I , as discussed in the section Decomposing the response . The concave FRET response of CheZ mutant cells is a consequence of the cooperative dePHOSphorylation of CheYp by CheZ : for small receptor activities , [Yp] is low , CheZ is mostly singly occupied by CheYp , and since the catalytic activity of CheYpCheZ , , is relatively small ( as compared to that of , ) , a given increase in must be balanced by a relatively large increase in and hence the FRET signal ; for higher , increases , CheZ becomes doubly occupied with CheYp , and since has a higher catalytic activity than CheYpCheZ , a given increase in receptor activity is balanced by a relatively small increase in . Indeed , if would be similar to , as Silversmith et al .propose [7] , the FRET response of the CheZ mutant cells would not be concave , and no good fit to the data of Vaknin and Berg [2] could be obtained .