TI - In wild-type cells , a small fraction of CheZ , of 10-20% , is bound to the receptor cluster , while the remainder diffuses freely through the cytoplasm . Figure 2b of Vaknin and Berg [2] shows the cyan signal , coming from CFP fused to CheZ , after the addition of attractant . . AB - This signal represents the spatial distribution of CheZ . The figure suggests that about 10-20% of CheZ is bound to the receptor cluster , with the remainder more or less homogeneously distributed in the cytoplasm . This estimate is consistent with that based on the known chemistry of CheZ binding to the receptor cluster . CheZ can be localized to the receptor cluster via binding to CheA , which is part of the receptor cluster . CheA exists in two forms , CheAs and CheAL , which can form the following dimers : CheALCheAL , CheALCheAs and CheALCheAs . The first two , CheALCheAL and CheALCheAs , have catalytic activity and can transfer phosphoryl groups to CheY [17] -[20] ; the third , the homodimer CheAsCheAs , does not have catalytic activity , but can bind CheZ . Earlier experiments suggest that CheZ binds selectively to CheAs[8] , [10] , [21] , although recent FRET experiments indicate that CheZ also binds to CheAL[9] . Following Lipkow [11] , we estimate that the number of CheAsCheAs homodimers is about 360 , while the number of CheZ dimers is about 1600 [22] . If we assume that CheZ predominantly binds CheAsCheAs , and that each of the CheAsCheAs homodimers strongly binds one CheZ dimer , we arrive at the estimate that about 20% of the CheZ dimers is bound to the cluster , consistent with the estimate based on the FRET data of Vaknin and Berg [2] . In wild-type cells , CheYp has a much higher affinity for CheZ bound to CheA than for CheZ freely diffusing in the cytoplasm . Figure 3a of Vaknin and Berg [2] shows that in non-stimulated cells containing wild-type CheZ , the total amount of [YpZ] in the cytoplasm roughly equals that of [YpZ] at the receptor cluster ; yet , as mentioned above , Figure 2b of Ref . [2] shows that the total amount of CheZ at the cluster is about 10-20% of that in the cytoplasm ; this means that CheZ bound to CheA at the receptor cluster has a higher affinity for CheYp than CheZ in the cytoplasm , as can also be seen directly from Figure 2d of Ref . [2] . The higher affinity could be due to a lower enzyme -subSTRate dissociation rate , or a higher enzyme -SUBstrate association rate . We assume that binding of CheZ to CheA increases the association rate . It is conceivable that CheA enhances the CheZ - CheYp association rate in a manner analogous to the gain of function mutations in CheZ studied by Silversmith et al .lt @@@@@ gt [7] : . . . . . . . CheA might relieve inhibition of the binding of CheYp to CheZ . A more speculative hypothesis is that CheA increases the CheZ - CheYp association rate because of the close physical proximity between CheA , where CheY is phosphorylaTED , and cluster-bound CheZ : a CheY molecule that has just been PHOSphorylated by a CheAp dimer at the cluster , can very rapidly bind cluster-bound CheZ ; in fact , if CheYp would be directly transferred from CheAp to CheZ , the association rate could even exceed the diffusion-limited rate . In wild-type cells , CheZ bound to CheA at the receptor cluster has a higher phosphatase activity than CheZ in the cytoplasm . The experiments of Wang and Matsumura [10] suggest that the interaction of CheZ with CheA enhances its dePHOSphorylating activity . This could either be due to a higher CheZ-CheYp association rate , or to a higher catalytic activity . We assume that binding of CheZ to CheA not only increases the CheZ-CheYp association rate , as discussed above , but also the catalytic activity of CheZ . In CheZ mutant cells , CheZ cannot bind to CheA at the cluster . CheZ in these cells has the same phosphatase activity and the same binding affinity for CheYp as CheZ in wild-type cells that is not bound to CheA at the cluster . As crystallographic data [23] and mutagenesis data [20] suggest , we assume that in the CheZ mutant protein only the domain that allows it to interact with CheA is affected ; the part that allows the CheZ mutant protein to interact with CheYp is thus assumed to be unaffected . This assumption is not critical for obtaining a good fit of our model to the data of Vaknin and Berg [2] . It is merely a simplifying assumption to reduce the number of free parameters . Indeed , it would be of interest to characterize the enzymatic activity of CheZ F98S--the CheZ mutant used by Vaknin and Berg [2] --since experiments by Silversmith et al .show that mutations far from the active site can , in fact , significantly change the enzymatic activity of CheZ [7] . For reasons of clarity , we first disregard the cooperativity in the phosphatase activity of CheZ . The CheZ mutant cells are thus described by the reactions of Eqs. 1-3 , while the wild-type cells are described by the reactions of Eqs. 1-2 , Eq. 3 for the reactions involving diffusive CheZ and the following reactions involving localized CheZ : . . . . . . . (8) Here , the total concentration of localized CheZ , , is low as compared to the total concentration of CheZ , . Furthermore , the association rate and the catalytic activity of localized CheZ , are high as compared to the corresponding rates and for diffusive CheZ . As we will show below , the critical parameters of this model are the fraction of CheZ bound to CheA at the receptor cluster , the ratio of the association rates and the ratio of the catalytic activities . The model presented here is similar to that of Lipkow [11] in that both assume that part of CheZ can bind the cluster . However , the models also differ in two important aspects : . . . . . . 1 ) in the model of Lipkow [11] , the binding of CheZ to CheA is conditional on the binding of CheZ to CheYp ; consequently , while in our model the bound fraction of CheZ is fairly constant in time , in the model of Lipkow [11] the amount of CheZ bound to the cluster depends upon the current stimulus level : . . . . . for instance , in her model , after the removal of attractant , CheZ moves from the cytoplasm to the cluster upon binding of CheYp ; 2 ) in the model of Lipkow [11] , the binding of one CheYpCheZ pair to a CheA homodimer , can nucleate the formation of oligomers of CheYpCheZ pairs at the cluster . However , as mentioned above , recent in vitro[7] , [16] and in vivo experiments [9] seem to disprove the idea of CheZ oligomerization . Our calculations reveal that CheZ oligomerization is not necessary ; the conditions listed above , are sufficient to explain the FRET data of Vaknin and Berg [2] . Moreover , the relative simplicity of our model makes it possible to elucidate the mechanism by which differential enzyme -SUBstrate binding affinity and differential catalytic activity can sharpen the response curve . Figures 4-6 show how the total amount of CheYpCheZ pairs and CheYp is affected by varying the critical parameters in this model : the fraction of CheZ bound to the cluster ( Figure 4 ) , the rate at which CheYp associates with CheZ at the cluster ( Figure 5 ) , and the catalytic rate of CheZ at the cluster ( Figure 6 ) ; the baseline parameters are given in Figure 4 . In all figures , the black line corresponds to CheZ mutant cells ; the red line corresponds to CheZ wild-type cells with the baseline parameter set ; the green and blue lines correspond to the results of the CheZ wild-type cells , where the parameter of interest is either increased or decreased ( see caption for parameter values ) . The black and red symbols correspond to the experimental results of Vaknin and Berg [2] , as described in section Decomposing the response ; the value of was , somewhat arbitrarily , taken to be , which means that is sigmoidal for CheZ wild-type cells and hyperbolic for CheZ mutant cells . The origin of the hyperbolic curve of the CheZ mutant cells is similar to that which underlies the response curves of the canonical model : , where initially , as increases from zero , is constant but then decreases as increases significantly ( see section Original Model ) . We will now discuss the origin of the sigmoidal curves of of the wild-type cells . Figures 4-6 show that the response curves of of wild-type cells effectively consist of two parts , corresponding to the binding of CheYp to cluster-bound CheZ and freely diffusive CheZ , respectively . When is low , a CheY molecule that has just been PHOSphorylated by a CheA dimer at the cluster , will most likely bind a CheZ dimer that is bound to the cluster because of the higher association rate between CheYp and cluster-bound CheZ , as compared to that between CheYp and freely diffusive CheZ : . Since cluster-bound CheZ has a high phosphatase activity , the concentration of CheYp and hence CheYp bound to CheZ will initially increase only slowly with . Nevertheless , at some point CheZ at the cluster will become saturated with CheYp . At this point . When is then increased further , a PHOSphorylated CheY molecule can no longer bind a cluster-bound CheZ dimer . It will then diffuse into the cytoplasm , where it can bind freely diffusive CheZ . Since the catalytic activity of CheZ in the cytoplasm is lower than that of CheZ bound to CheA at the cluster , [YpZ] and will now quickly rise . This combination of differential affinity and differential catalytic activity thus provides a generic mechanism for enhancing the sharpness of the response . We can now understand the effect of varying the critical parameters in this model . As the fraction of CheZ that is bound to the cluster increases ( from green to red to blue in Figure 4 ) , the amount of CheYp needed to saturate cluster-bound CheZ increases , leading to a shift of the inflection point in to higher values of . However , while increasing the fraction of cluster-bound CheZ shifts the inflection point to higher values of , it does not significantly change the initial slope of , nor does it change the slope after the inflection point : these slopes are determined by the catalytic activities of cluster-bound CheZ and freely diffusive CheZ , and , respectively . This can be seen in Figure 6 : as the catalytic activity of is increased ( from blue to red to green ) , the initial slope of decreases . Please also note that since the slope of after the inflection point is determined by parameters of freely diffusive CheZ , it is similar to the initial slope of of the CheZ mutant cells , which indeed only contain freely diffusive CheZ , exhibiting the same phosphatase activity as diffusive CheZ in wild-type cells . Figure 5 illustrates the importance of the association rate . As the rate of association between CheYp and cluster-bound CheZ decreases ( from red to blue to green ) , the response curve of CheZ cells moves towards that of the CheZ mutant cells . The reason is that as the rate of association between CheYp and cluster-bound CheZ is lowered , it becomes more likely that a phosphorylaTED CheY molecule diffuses into the cytoplasm , where it will be dePHOSphorylated by freely diffusive CheZ with a lower catalytic activity . The differential-affinity-and-activity model is able to explain the measured difference between the response curves for the CheZ mutant cells and the CheZ wild-type cells . However , while the response curves of Vaknin and Berg [2] can be reproduced by the model , this is not the only constraint . As discussed above , both wild-type and CheZ mutant cells should be able to chemotax [12] . This means that the model should give CheYp concentrations between 1 and 5 uM for both strains in the non-stimulated state [14] . As can be seen from the fit used in Figures 4-6 , in the CheZ mutant , the CheYp concentration is 8 uM in the non-stimulated state , which is well outside this range . This fit can , however , be improved by taking into account the effect of cooperativity in the phosphatase reactions , which we have neglected thus far in the differential-affinity-and-activity model . The reactions of diffusive CheZ , both in the wild-type cells and in the CheZ mutants cells , are given by Eqs. 3 and 5 , while the reactions involving CheZ localized at the receptor cluster in wild-type cells are given by Eq. 8 in combination with . (9) As before , we assume that both the affinity to CheYp and the phosphatase activity of CheZ are enhanced when CheZ is localized to CheA at the receptor cluster . This means that the association rates and are much larger than the corresponding association rates for cytosolic CheZ , and that the catalytic activity is larger than the catalytic activity for cytosolic CheZ . Figure 7 shows and for CheZ wild-type cells and CheZ mutant cells [2] . In combination with a response curve for versus . [Serine] with , the four dose-response curves in Figures 5a and 5c of Ref . [2] are reproduced . Comparing Figure 7 with Figures 4-6 of the simplified differential-affinity-and-activity model shows that the cooperative dependence of the phosphatase activity on CheYp concentration does not dramatically affect the dose-response curves , a conclusion that was also reached by Sourjik and Berg [14] . Indeed , in this model it is possible to obtain a good fit to the data [2] while assuming that the catalytic activity of CheZ is independent of the number of bound CheYp molecules , as suggested by the in vitro observations of Silversmith et al .lt @@@@@ gt [7] ( data not shown ) ; the critical ingredients of this model are that the binding affinity and catalytic activity of cluster-bound CheZ are higher than those of freely diffusive CheZ . As for the model without CheZ cooperativity , is in agreement with experiment , both for CheZ wild-type and CheZ mutant cells . Moreover , the response curve of the CheZ wild-type cells agrees with experiment in the sense that the concentration of CheYp equals 2 uM in the non-stimulated state , which is within the working range of the motor . The concentration of CheYp in the CheZ mutant cells in their non-stimulated state is around 5 uM , which is lower than that in the simplified differential-affinity-and-activity model , but still at the high end of the working range of the motor .