- Email: [email protected]
Two-state model for bacterial chemoreceptor proteins
*I. hid. Rd. (1984) 176, 349-367
Two-state
Model
for Bacterial
Chemoreceptor
Proteins
The Role of Multiple Methylation SHO ASAKURA AND HAJIME
HONDA
institute of Molecular Biology, Xchool of Science Nagoya University, Xagoya 464, Japan (Received
19 December 1983)
To help understand the bacterial chemotactic response of excitation and adaptation. we propose a simple two-state model for receptor proteins (methylaccepting chemotaxis proteins), in the light of evidence that they undergo multiple methylation in a preferred order. The model includes the following assumptions. (I) The receptor protein is in rapid equilibrium between two conformations, S and T, and the equilibrium shifts towards the T form as the number of methyl groups increases. (2) Attractants bind to the S form of the receptor. repellents bind to the T form, and both classes of ligand shift the S/T equilibrium according to the mass-action law. (3) The S form of the receptor accepts methyl groups one by one in a definite order. while the T form releases the methyl groups in the reverse order. Methylation and demethylation are slow reactions, and changes in the total number of methyl groups lag behind shifts in the S/T equilibrium. (4) The pattern of bacterial swimming at any moment, is determined by the partition of the receptor between the two conformations. with tumbling frequency being a monotonically increasing function of the total T fraction of the receptor. This model shows that, if the receptor satisfies two sets of relationships imposed on its equilibrium and kinetic constants, it can maintain the steady-state total T fraction essentially constant over a broad range of ligand concentration, enabling cells to adapt to large changes in chemical environment. A stepwise change in ligand concentration leads to a rapid change in the total T fraction (excitation). followed by a slow relaxation process (adaptation). Computer simulations have been made of the whole response process, employing a receptor with six methylation sites per molecule and assuming simple sets of parameters. The results are in general agreement with published data on receptor methylation, as well as with a variety of observations of bacterial chemoresponsr. Multiple methylation of the receptor proves to be necessary for the cells to respond sensitively to environmental changes.
1. Introduction lSacat,erial cells migrate through spatial concentration gradients of attractants and repellents. Typical attractants include serine, aspartate, maltose, rihosr and galactose: repellents include leucine, indole and ?ji2+. During chemot,actic migration, cells detect, via specific chemoreceptors, temporal changes in concentration of ligands, and they respond by altering their pattern of swimming t’hrough regulation of the sense of flagellar rotation. Counterclockwise rotation of
350
S. ASAKL~RA ASI) H. HONIIA
the flagella (as viewed from their distal end) produces smooth swimming. a11c1 clockwise rotation causes tumbling. The cells suppress tumbling for a periocl of’ time in response to attractant addition and repellent removal: conversely. the! enhance tumbling in response to attractant removal and repellent addition. Thrsr initial responses to environmental changes are followed by a period of adaptation in which the cells return to their pre-stimulus pattern of swimming. A remarkable feature of the phenomenon is that bacterial cells are able to adapt to a broad concentration range of chemical environments, but also to respond sensitively to small changes in them (for reviews. see Macnab. 1978: Springer r~f nl.. 1979: Koshland, 1980,198l; Boyd Ilr. Simon? 1982). In the bacterial chemoresponse of excitation and adapt’ation. a set of chemotaxis membrane-associated proteins, designated i.he methyl-accepting proteins or M(:Ps. play a central role (Kort’ c!t al., 1975; Springer et f~1.. 1979). Three MCI’s, all about 60,000 M,. have been ident,ified in Escherichia colt' (Springer et al., 1977; Silverman $ Simon, 1977; Kondoh rt nl., 1979; Koiwai et ul.. 1980). MCPT is necessary for response to serine. leucine and indole; MCPTl for response to aspartate, maltose and Xi2+: MCPIII for response to ribose and galactose. Methyl groups are transferred from S-adenosglmethionine to glutamyl residues in the MCI’s by a specific methyl-transferase. and the result’ing glutamylmethyl esters are hydrolyzed by a specific esterase (Kleene rt ul., 1977: Springer & Koshland, 1977; Coy et al.. 1977: Stock & Koshland. 197X). The MCPs art‘ multiply methylated (Boyd 8: Simon! 1980; Chelsky & Dahlquist. 1980: DeFranc*o & Koshland. 1980: Engstriim CyrHazelbauer. 1980): they have as many as six glutamyl residues per molecule that) are methylated in a preferred order (Kehry Q Dahlquist, 1982; Springer et al., 1982). The methyl-accepting glutamyl residue5 have been identified in the primary structures of MCPs (Krikos rt aZ.. 1983; Russo & Koshland, 1983). A single cell cont’ains more than 1000 molecules each of MCI’1 and MCPTT (Hayashi et ab., 1979; Stock & Koshland, 1981a). which account for, 900/:, of t,he t,otaI MCPs (Hazelbauer et nl.. 1981). L\:h en cells are in an isotropic chemical environment, the MCPs maintain a constant level of methylation, which in the presence of at’tractants (or repellents) is higher (or lower) than in their absence (Coy et al., 1977: Springer rt nl.. 1979). A wealth of evidence shows that MCPI and MCPTI are the serine and asparta,te receptors, respectively (Clarke & Koshland, 1979: Wang Br Koshland. 1980). and a mechanism for control rtf receptor methylation by ligand binding has been proposed by Stock B Koshland (1981a). Wit,h this experimenta,] background. we present here a simple two-si.atjcx model for the chemoreceptor protein. t’o throw light on its function at the initial step of sensory transduction in bact,eria. Goldbrter k Koshland (1982) have reportSed detailed analyses of a series of models for the receptor protein. Tttc present model is an altjernativr. 2. Results and Discussion (a) Model
The model is based on the following assumptions. (1) The receptor protein is in rapid equilibrium bet,ween two conformations. which are designated S and T because of their controlling effects on t.hc
MODEL
FOR
BACTERIAL
CHEMORECEPTORS
35 I
smooth/tumbly behavior of the motor organelles (see assumption (4)). The equilibrium shifts towards the T form as the number of methyl groups is increased. If the receptor protein molecule has a total of n sites that accept methyl groups one by one in a definite order, it partitions the methyl groups among (n+ 1) molecular species. The fraction of protein molecules present as the species having i methyl groups is denoted by Pi: $Opi
=
(1)
l.
Let Lp stand for the S/T equilibrium constant, in the absence of ligand, for the species having i methyl groups. Then, by assumption: L; > Ly >
> L,o .
(2)
(2) Attractants bind only to the S form of the receptor, and repellents bind only to the T form. Affinity between an attractant (or repellent) and the S (or T) form of t’he receptor is independent of the number of methyl groups, the dissociation constant being referred to as K, (or KR). Then, the mass-action law leads to the following relationships for any concentration of the attractant [A] or of the repellent [R] : (Al/K,
= [$,~A]/[&,]
= [S,.A]/[S,]
= . . . = lS;A]/[S,],
[RI/K, = [T,~RI/[R,I = [T,.RI/[T,I =
. = P’,~W[Tnl.
(3)
Normalized concentrations a = [AI/K, and /l = [RI/K, will be used hereafter in place of [A] and [RI. The sum of the [S,.A] and [Si] fractions is denoted by Si, and the sum of the [T,.R] and [Ti] fractions is designated Ti: Si + Ti = Pi.
(4)
Then, the Si/Ti ratios, referred to as Li terms, can be expressed as: ,1+ci Li = Si/Ti = L il+p
(i = 0, 1,.
) n).
Changes in ligand concentration lead to instantaneous changes in Li terms according to equation (5). Note t’hat, at any concentration of the attractant (or repellent), its overall affinity towards the receptor decreases rapidly with an increase (or decrease) in the number of methyl groups, since: L, > L, >
. > L,.
(6)
The sum of the Ti (or Si) values over all molecular species is denoted simply by T (or 8): T=
i Ti. i=O
Given the inter-species distribut’ion from:
s=
tsi
(T+s
= 1).
{Pi} at any moment, we can calculate T and S
T = t Pi/(1 +L,) and S = i P,L,/( 1 + Li), i=O
(‘i)
i=O
i=O
(8)
Xr’ and fractional
S. AS.-ZKI-RA AND H. HONDA receptor
occupancy ITA
zz
from :
s”-
1 +!x’
y R = 7’2 1+P
(3) The t)wo conformations of the receptor are recyyizrtl as being dist inc.t t)>the meth?il-t,ransferase and the esterase. The former enzyme catalyzes the transfer of methyl groups to each Si fraction (and not to ‘I‘, fractions). and the Iat trr enzyme cat.alyzes the removal of methyl groups from taac.1~Ti fraction (and IIOI from Si fract,ions), as depicted in the following scheme:
(IO]
where the sub-species of Si (or Ti) with and wit,hout bound l&and are assumed t)o be equally susceptible to methylation (or demcthylation). The methylation and demethylation are slow react.ions, and shifts in the inter-spec+s distribution lag behind changes in Li values. I)enot,ing the apparent rate c-onst,antb for met,h,ylation and demethylation at f,he ith step by CZ( and hi. then the rittcs equations governing the inter-species distribution are: I’” = -u,S,+h,7’,. Pi = -Ixi+l
Si+h;+,Ti+,
I’, = a, s,
, - h, T,
+Q\‘~
1~-h,T,.
ill)
These interdependent first-order differential equat)ions (aan h(b transformed into H independent &h-order differential equations. (4) The pattern of bacterial swimming at an!’ moment is tlet~erm~netf Ir)>. the partiCon of the receptor protein between thus t\vo (*onformations. a-ith 1he frequency of tumbling or the bias of flagellar rotation (the fraction of time spent rotating clockwise) being a monotonically increasing function of the total ‘I fraction, T. of the receptor. (To specify the details of t,his function. w-e nould hart* to elucidate the nature of the signal that is (Lontrolling the bias of flagellat motors.) This receptor model has elements in cbommon \vit,h thoscb proposed 1)~ lioshlanti (1981) and Stock & Koshland (1981h). which, however. have not been analyzrd mathematically. Tt should be mentioned that, the present model arose from caonsideration of the observed changes in switnming pattern of non-adaptive mutants. Cells missing the e&rave exhibit continuous tumbling. but they can hr made to swim smoothly by addition of attract)ants (Parkinson. 1977). (‘ells missing the transferase are usually smooth swimming hut can tumble frequrnt,l>. in the presence of repellents (Springer & Koshland. 1977: Parkinson & Revcllo. 1978: Coy et al.. 1979). These facts have led us t,o assume t’hat (1) the signal for
MODEL
FOR
BACTERIAL
353
CHEMORECEPTORR
tumbling is generated by the receptor regardless of the methylation level and the occupancy, and (2) attractants and repellents cause allosteric transitions in the receptor protein. According to this model, the changes in swimming pattern of a non-adaptive mutant with ligand concentration are governed simply by the massaction law, as the receptor is frozen either in the state that P, = 1 (for which it is assumed t,hat Lz K 1; see below) or in the state that P, = 1 (Li >>1). [Jnder steady-state conditions, equations (11) reduce to:
pi = ai/bi= T,/AS-~
(i = 1) 2,.
, n).
(12)
Combination of equations (4), (5) and (12) yields the steady-state’ Pi values as functions of ligand concentration. As the concentration of an attractant (or repellent) is increased, the steady-state inter-species distribution shifts towards the right-hand (or left-hand) side of the scheme of equation (lo), and these shifts are accompanied by a lowering of the overall afinity between the ligand and the receptor, as stated earlier. So, any ligand has its affinity for the receptor decreased in the steadyas the concentration is increased, leading to negative co-operativity stat’e binding. The co-operativity vanishes if methylation/demethylation is blocked. It will be seen shortly that, according to the model, the receptor can maintain its steady-state T fraction essentially constant over a broad range of attractant and repellent concentrations, provided the following two sets of conditions are satisfied simultaneously. (1) The S/T equilibria of the unmethylated and the fully methylated species heavily favor the S and T conformations, respectively: L~>>landL~<
(13) by the same ratio
pi = I* (constant) for all i. Under condition
of the rate
(14)
(2), equations (12) reduce to:
p = T,/S, = T,/S, = . . .= T,/S,-,
= (T-To)/@-S,),
(15)
where, under condition (l), both To and S, are virtually zero within a broad range of repellent and attractant concentrations. In this range, therefore, the receptor can maintain a constant steady-state T level at F = p/( 1 +p). This is an important property of this model that accounts for the capability of bacteria to adapt to large changes in ligand concentration (see below). Consider changes in Li, Pi and T brought about by a stepwise change in concentration from a( 1) to ~((2) at time t = 0. Steady-state values for a variable z before and after the change are referred to as z(1) and z(2), respectively, and the value immediately after the change is denoted by z(0). Then:
Li(0) = L,(2) and P,(O) = P,(l)
(i = 0, 1,.
) n)
(16’)
X-1
S. ASAKVRA
AND
H. HONI)A
and insertion of these relations in equation (8) yields: 7’(O) = i Pi(l)/{1
+I&)).
( 17)
i=O
If the steady-state Pi values have hrrn determined as functions c)f‘ t I~(, concentration of ligand. it is easy t,o calculate the T(O) valuc~ for any stepwisca stimulus from equation (17). The difference:
T(O)-T(2) = ~{Pi(l)-Pi(2)j:{l
+I&)).
!IH)
may be used as a measure of the receptor excitation evoked by a given stepwlst‘ stimulus. In order tha,t a stepwise stimulus should evoke strong excit#ation. therv should be a large difference in the steady-stat,e inter-species distribution hefort, and after the stimulus. Tn other words, t’he stead\--stat,e inter-species distributjion should strongly depend upon the concentration of ligand in order t)o make the receptor highly excit,able. This requirement is fillfilled hy virt,ue of multiple methylation (see below). The relaxatjion from 7’(O) to T(d). corresponding to adaptation, is governed by rate equations (1 I). which n-e solved hy comput’er IO simulate the process. The results are presented in the following section. (h) A nnlyxin Here we deal with t,he model numerically, employing the recept,or wit)h I/ = 6 and assuming simple set,s of parameters in acvorda,ncne with equations (1.3) and (14). The first example is: (‘asp A:
i
0
1
2
log I$
5.5
1.r,
3
4
;i
ti
I .o 0.5
0
-0.5
-4.5
p = 0.1: ui = I s- ’ and bi = IO s - ’ for all i values. Here. the values for I,: to IAt are distributed in simplicity. The use of the value 0.1 for p will parameter values, equations (5) and (1.5) lead partition of the receptor among the Lci and Ti attractant) and repellent. with .r = (1 +cr)j( 1 +/I): 0
i 6.
1p5r-2
7:- 1()-5,r-J
I 1OS- l l()-O.“,r-2
a geometric series merely fi)t be disrussed later. For thrsv to the following s&ad)--state species in an environment of
1
ti 1 t) 3 10 I oo.5.r 10 -()‘s,r2 IO 2,r3 IO- -.5 .x 4 1()-o’“/. ,()-l.s,r.2 lo- 3x2 ,f ’ 1
It is evident that, t,he steady-state statistical wrights of T, and 8, are virt nall~ zero over a broad range of /I and 2 values, although T, -+ 1 as /3 -+ x and S, + 1 as a -+ CD. From t’his sequence. it is easy t,o cal(ulatrl the inter-species distribution and the T value under steady&atr conditions. as functions of t,he concentration of liqand (Fig. 1). The calculated dependence of the inter-species distribution upon concentration is in general agreement with data obtained by gel electrophoretic resolution of the molecular species (Royd & Simon. 1980: (‘helsky & IIahlquist. 1980; DeFranco &, Koshland. 1980). With in the rang? of ligand concentration
MODEL
I’Olt
BA(‘TEKlAL
(‘HEMORE(‘EI’TORS
Log(lfa)
FIG. I. Steady-state properties of the receptor in case A, plotted against log (1 +G() and log (1 +/I). The fractions of the 7 possible molecular species. Pi values (thin continuous lines, the number on each c~arve indicating the number of methyl groups), the extent of met~hylation ,XiPJfi (dotted line). and the 7 Iwel (thick continuous line) are shown.
given in Figure 1, the T curve is approximately horizontal, with an average value T = I /I 1. More specifically, the T value remains constant, 0.0909+0.0009, over the range of IX < 103’5 and /I < 102’5. A deviation of T as small as 1y& would presumably not cause any observable change in the bias of flagellar motors, in which case cells can adapt to as large a range as a < 103.5 and fl < 102’5. This range extends further as Lg is decreased from 10-4.5 and Li is increased from 105,5. The fract’ional receptor occupancy (eyns (9)) at steady-state, within the adaptive range of ligand concentration in case A, can be approximated by: y,=--
10
c1
11 1+a’
1 yR=llI+p.
B
Therefore, the steady-state fractional occupancy appears to increase in a hyperbolic manner with ligand concentration, approaching a limiting value smaller t’han unity, as if a part of the receptor were sequestered. This behavior, is due to t’he negatively co-operat’ive which may be termed pseudo-saturation, ligand binding mediated by methylation/demethylation. Measurements of t.hr pseudo-saturation curves yield the dissociation constants K, and K,. It’ is t.o he expected that, when this procedure is used to compare a wild-type strain and a non-adaptive mutant strain lacking the transferase (or esterase), the same K, (or KR) value will be obtained from them. The rate of methanol production, which was first measured by Toews cut rcl. (1979), corresponds to equation (20):
J = xbi7’i.
S. ASAKL’ILA
AND
0 Loqiltp)
H.
HO91)A
I 2
1 I
I 3
L.3qt1taI
Fro. 2. Changes in I[’ upon applying stepwiar changes in ligand concentration continuous and broken lines represent the T(O) values when ligand concentration decreased to zero, respectively. Thick continuous line: the steady-state T level.
in case A. The thm is increased from and
In case A, where we assume bi = 6 (constant) for all i values. tjhe steady-state J are constant, at the value values over the adaptive range of ligand concentration bT. This is consistent with the observation by Toews rt al. (I 979) that the steady state rate of methanol production in the presence of high concentrations of attractants was virtually equal to that in their absence. However, it may be an over-simplification to allot the same value to t’he b, terms, alt’hough ai/bi ratios at all methylation steps must be constant. Suppose that h, is larger t’han t,he other bi t,erms, while ai/bi = 0.1 for all i values as in case A. Then, under steady-state conditions, J will have a maximum value at log (I + 2) = 0.75 (x = 4%), because T, is at’ its maximum at t’his concentration. Tncidentally, the steady-state values for T, and P, have maxima at CLvalues close to each other (Fig. 1). Transient changes in T upon applying stepwise changes in ligand concentration are shown in Figure 2. Increases in ~1and decreases in /I lead to negative values for T(O)-T(2) (corresponding to a decrease in tumbling frequency). decreases in r and increases in /I lead to positive values (enhanced tumbling frequency), and large concentration differences give rise to large values for IT(O)-T(2)/. These results suggest that the model may work well to describe excitation. Computer simulations have been made of the T(O) -+ T(2) process in case A, where ai= Is-’ and hi= 19s -I for all i values. The time-scale should be considered arbitrary. Although the assumed set of rate constants is not necessarily realistic, the results make clear the general transient properties of the receptor. It is seen from Figure 3 that the excitation evoked by addition of a high concentration of an attractant (or repellent) relaxes considerably more slowly (or rapidly) than that evoked by its removal, in agreement with the well-known observations by Macnab & Koshland (1972) and by Berg c1:Tedesco (1975). The
0.
I-
MODEL
FOR
HA(:TERlAL
-
Attractant addition
CHEMORECEPTOKS
Xii
Repellent
T
()Removal
T
Add’hcn
0.I 5-
(I-
l 0
I
I
I
I
4
2
FIG 3. Relaxation processes of 7’ in case A. Ligand addition: stepwise increase in a or /I from 0 to 99 rep at f = 0. and removal: decwasr from 99 to 0. Time was divided into 042 s intervals for soloing (I 1) hy
computer.
relaxat~ion may he characterized in terms of the half-time t1,2 (the time required for (T-T(2)}/{?‘(0)Y’(2)) to reach the value l/S). The tIiz values calculated for various kinds and amplitudes of stepwise stimuli are shown in Figure 4. The four curves appear to approach limiting levels with increases in stimulus amplitude. r-
I-
J; 19
i
--\
\
I
\
‘\ c)-
I
I
2
I
---f-k-
Log (I tp,
FIG. 1. The I,,, values ralculated linrs) and for decrrases tn
(continuous
for zero
‘---- _ LogCltCY)
stepwise (hrokrn
increases lines)
in case
in lipand A.
concentration
hm
7rw.1
3%
S. ASAKI‘ltA
AND
H.
HOSIIA
The ratio between the limiting levels for addition and removal of an attractant IS about 12. This ratio increases further as the parameter p is decreased from 0.1. times for addiGon of’ high Berg &, Tedesco (1975) showed that, transition c:oncent.rations of an attractant were 100 times as long as those for its remova,l. III view of this. the true ,Dvalue may be smaller than 0.1. In F’igurcs 4, the four (*urvt+ start from the same height, of the ordinate, t,/, = (M4 seconds. This implies that excitations evoked by very sma.11amplitude stimuli of attracat’ant and repellent relax at a finit,e rate (see below). The magnitude of the response of a receptor caan be c~xprrssc~tlin ttarrns of’ A = i (7’-7’(2)]dt. 0
(“I)
which may be positive or negative depending upon the nature of a given stimulus. Figure 5 shows the A values calculated for various kinds and amplitudes of stimuli in ease A. The two sets of A curves in the first and fourth quadrants and those in the second and third quadrants are approximately mirror images. respectively. Let A,, stand for response to a stepwise stimulus, c1(1) -+ ~(2) or /?(I) -+ p(2). and let 6 be:
0.6
I
FIG. 5. Thr integrated response (A) values wlculated for case A. The 3 wntinuous rurvc~ in thr 4th quadrant (lower right) represent the A values for stepwise increases in 1 from 0. 9 and 99 to thts indicat,ed values. The curves in the first quadrant (upper right) represent the values for decreases in z from the indicated values to 0, 9 and 99. The 2 sets of curves in the second (upper left) and 3rd (lower left) quadrants, obtained for repellent stimuli. have similar meanings Integration of eqn (21) was function of st,opped when ld!J’/dtl became less than 2 x IO- ’ s -I The dotted line indicates a hyperbolic a with the half-maximum at c( = I, which represents the steady-state receptor occupanyv in t,hr adaptive range of &and concentration.
MODEL
FOR
BACTERIAL
CHEMORECEPTORR
350
Fro. 6. Changes in 7’ brought about by addition of an attractant (CL= 0 + 99) at t = 0, followed by its removal (99 + 0) at t = 0.5, 1.0 and 1.5 s, in case A. In each response curve, the area of the positivr part is larger than that of the negative part; however. the difference is smaller than 5O;,.
Then, 6 < 0.01 in the cc-region, and 6 < 0.07 in the P-region. Furthermore, the A curves in each quadrant can be superimposed closely upon each other by translation along the ordinate. Let 6’ stand for: (22b) Then, 6’ < 0.002 for increases in a, and 6’ < O-02 for decreases in b. Thus, A values and decrements in ligand concentration are for consecutive increments approximately additive. Figure 6 shows responses to attractant pulses of various widths. Each T curve consists of negative and positive parts (as measured from the steady-state level), the areas of the two parts being approximately equal (6 < 0.05). These and many other examples indicate that responses are approximately additive regardless of the intervals of time between successive st,imuli. The additivity of receptor responses reminds us of the additivity of bacterial responses as observed by measurements of recovery times (Spudich &, Koshland, 1975) and of transition times (Berg & Tedesco, 1975). However, before discussing the relationship bet’ween the experimental and theoretical results: we have to specify the functional T-dependence of the frequency of tumbling or the bias of flagellar motors. It is seen from Figure 5 that response to attractant, addition (the outermost continuous curve in the fourth quadrant) is not proportional to the steady-state receptor occupancy (the dotted line). We conclude therefore that, in the present receptor model, the additivity of responses is not based upon the proportionality between responses and changes in occupancy. Goldbeter & Koshland (1982) h ave also argued that the property of
additivity can lie dissociated from the l~ro~~ot~tionulit~~ l)rtween responses itrlti changes in occupancy. For very small stepwise stimuli, receptor responses are proportional to stimulus amplitude. and arc additive (Fig. 7). Lye obtain a value fbr the ~)roportionalit.!~ constant dlAl/dcr or dA/dj of 0.14 seconds, which tnay be regarded as the resl~msc sensitivity of the recept,or (case .A) at low concentration of ligand. Transient changes in 7’ upon applying very small stepwise stimuli are also proportional IO stimulus amplitudes (Fig. 7). This means that attractant and repellent stimuli of’ very small amplitudes evoke excitations that relax at a finite rate. The same conclusion has been drawn from the calculat’ion of t 1,2 (Fig. 1). These results are in accordance with some aspe& of data on impulse responses of wild-type ~~11s. reported by Block ef nl. (1982), although the existence of a stimulus threshold for evoking excitation cannot be explained by the present model. LVe should mention at t’his point that a simplitied version of t,hc receptor \rith II = 1 (L,O >> 1 and I,: << 1) can maintain a constant stead!--state T level over a broad range of l&and concentration. When, for example. /$ = 10s’“. I,? = 10 m4’i and p1 = 0.1, the steady-state T value is kept constant (0.0909 f 04009) ovet x < 103’5 and p < 102’5. as in case A. In this range of ligand concentratioii. however. the steady-state inter-species distribution remains virtually caonstant changes in 7 (PO = 0.9009 - 0.9092 and P1 = 04991 - 0.0908). and transient upon applying large st~epwise stimuli are negligible as compared to the stradystate level. That, is. the YI = 1 receptor (*annot work to excite the cells. \l’t. calculated A values for the receptor with YI = 3. assuming 1,: = 105’5. Ly = 10“5. At IOM z,; = 10-“‘5. Li = 10-4’5. )I CL-= 1 b‘i-l and hi = 10si - ’ for all i valnrs.
MODELFORBACTERIALCHEMORECEPTORR
XI
concentration of ligand, the response sensitivity of this receptor is 4.0-fold smaller than that in case A. It turns out from these comparisons that, in receptor A, the five molecular species, P, to P,, are co-operating to increase the response sensitivity within a range of (x < 103” and p < lo*“. where the steady-state T level is kept constant by the existence of the two extreme species. In conclusion. multiple methylation of the receptor plays a dual role: it allows the cells to adapt to large changes in ligand concentration, and it also increases their excitation sensitivity to small changes. For the second example of the n = 6 receptor, we assume the following set of parameters: Case B:
i log
0
1
2
45
3
6
L; 5.5 1.0 0.75 0.5 0.25 0 -4.5
a,=1s-’
and bi = 10 s-l
for all i values,
more closely than in case A. It is where the values for Ly to Lt are distributed seen from Figure 8(a) that the steady-state Pi curves in case B are located more closely than in A. The steady-state T levels in the two cases coincide over a range of CI < lo3 and p < 103. The A curves calculated for the two cases are compared in Figure 8(b). At low concentration of ligand, the response sensitivity of receptor H is 1.7.fold higher than that of A, while the sensitivity of receptor B decreases wit’h ligand concentration more rapidly than that of $. So, the former receptor is not) necessarily superior to the latter. Also, in case B, responses are approximately additive, although the response to attractant addition is not proportional to the steady-state occupancy (Fig. 8(b)). The third example is: Case C:
i log L;
0
1
2
3
5.5 1.25 1.0 0.75
ai = 1 s-l
and bi = 10 s-l
4
5
6
- 1.25 - 1.5 -4.5 for all i values,
where the L:/Lz ratio is assumed to be large. This non-uniform distribution of Li values gives rise to large P, values at intermediate concentrations of an attractant under steady-state conditions (Fig. 9(a)), leading to a biphasic A curve for its addition (Fig. 9(b)). In this case also, the steady-state T level remains constant over a < 103” and /? < lo*“, where the steady-state occupancy is simply hyperbolic. It is well-known that bacteria respond to increasing concentrations of serine in a biphasic manner (Springer et al., 1977; Hedblom & Adler, 1980). To interpret this phenomenon, it is usually assumed that the serine receptor has two sites with different dissociation constants. However, the above example suggests t’hat even a single binding site may, given the appropriate characteristics of t’he multiple methylation system, produce the biphasic response. The model can be applied to simulate receptor responses during continuous changes in ligand concentration. Figure 10(a) shows the response of receptor A to a sinusoidal change in attractant concentration of the form: a = 0.5{1-cos(27$)},
where f = 517 s-l.
(23)
T
o.!
Log iita)
Log (I cp, (0)
The calculated 7’ c~~rve is periodic and becomes stationary a fiw periods after the initiation of stimulus. The stationary waves rousist of tlw psitivr and negativr~ half-periods (as measured from the steady-&d Iwrl) that have the same area hut different amplitudes. Figure IO(b) is a tloublr logarithmic* plot of t,ht, normalized
MODEL
FOR
BACTERIAL
CHEMORECEPTORS
I*C )r
------A T
o-c:
\
Log(I+P)
Log (Ita)
Log cutp, (b)
FIG. 9. (a) Steady-state properties of the receptor in case C. For an explanation, see Fig. 1. (b) The IAJ curves calculated for attractant addition and repellent removal in case C (continuous lines). The broken lines represent the [A\ curves calculated when Li = 1Oo’5 and Lt = 10°‘25, with the other parameters being unchanged from case C.
mean stationary amplitude of the response as a function of frequency f (Bode plot). The results resemble the amplitude gain of a first-order high-pass filter. Block et al. (1983) have measured bacterial responses to exponentiated sine waves of an attractant at the low frequency side, and have found that the response was a first-order high-pass process. The combination of the experimental and theoretical results implies that the rate of bacterial response to a gradual change in ligand concentration is determined by receptor methylation.
304
S. ASAKIJKA
0
AND
H. HOSI)A
2
4 f, (0)
1
I-
0.01
I 0.01
I 0.1 f lb)
FIG. 10. (a) Change in T of receptor A in response to a sinusoidal change in at,tractant concentration given by eqn (23). The stimulus starts at t = 0. Abscissa: time multiplied by frequencyf = 5/7 SC’. The response wave becomes stationary a few periods after the initiation of stimulus. The mean amplitude of the stationary wave tends to 0.052 at high frequency. which coincides with the mean of the values for T(O)-Z’(2) for 2 independent stepwise changes, 0.5 + 0 and 0.5 + 1. (b) A double logarithmic plot of the normalized mean stationary amplitude as a function of f. The continuous line indicates the amplitude gain of a first-order high-pass filter.
Bacterial cells undergo thermotaxis (Adler, 1976; Maeda et al.. absence of ligand, they suppress (or enhance) tumbling frequency time in response to temperature increase (or decrease), and adapt. evidence shows that the thermal stimulus is detected mainly receptor and marginally by the aspartate receptor (Maeda & Imae, & Tmae, 1984). On these grounds, we assume that the equilibrium
1976). In the for a period of Experimental by the serine 1979; Mizuno constants, Lf’.
MODEL
FOR
I o
2
BACTERIAL
I I Log (I +p,
CHEMORECEPTORS
0
1 I
I 2
Log(lfa)
FIG. 11. Thermal excitation of a receptor that has the 2 sets of parameters, of cases A and R, at a given higher and lower temperature, when it is subjected to temperature jumps between the 2 values at various ligand concentrations. The thin continuous and broken lines indicate the T values Thick continuous line: the immediately after temperature increases and decreases, respectively. steady-state 2’ level. The 2 transient T curves intersect the steady-state curve at different points.
for each of these receptors depend upon temperature, while the parameter p is independent of temperature. If this assumption is correct, a temperature jump in the presence or absence of ligand would bring about instantaneous changes in L, (eqns (5)) for a given receptor, leading to a transient change in T. If two sets of Lp values of the receptor at two different temperatures are given, transient changes in T upon applying temperature jumps between the two values at various ligand concentrations can be calculated from equation (17). Figure 11 shows the results obtained for a receptor that has the two sets of parameters, of cases A and B, at a given higher and lower temperature, respectively. The thermal excitation is inverted as attractant concentration increases. This result is in agreement with the finding by Mizuno & Imae (1984) that mutant cells lacking the serine receptor protein exhibited inverted thermoresponse when they had become adapted to intermediate concentrations of aspartate. Therefore, it is likely that the distribution of Lp values in the aspartate receptor becomes broader with a rise in temperature. The present model may apply also to bacterial responses to weak acids (Kihara & Macnab, 1981; Slonczewski et al., 1982) and to glycerol and ethylene glycol (Oosawa & Imae, 1983). The nature of the signal controlling the bias of flagellar motors remains obscure. The elucidation of its nature is awaited for developing the model by determination of the functional dependence of the bias of motors upon T. It is probable that the serine, aspartate and ribose receptors are controlling motors by means of a common signal (such as a specific small molecule or ion). If so, the bias of motors may possibly depend upon “T” with respect to the total receptor, rather than for individuals, and the dependence around the steady-state in the absence of ligand is supposed to be very sharp, because ribose addition excites
XC
S. ASAKl:KA
AXI)
H. HONDA
cells to the full extent, whereas the ribose receptor (MCIPIII) amounts to onI> 10% of the total. From t,hese considerations, it is to be expected that t,herr is ii second mechanism for signal amplification in the bacterial chemoresponse. WI: thank Drs \‘. Imae, S. Kobayashi, H. Hayashi and I’rofcssor 11’. Oosawa 01’ this Institute for stimulating discussions. Thanks are also due to Prof&saor R. M. Macnab (Yale University) for critically reading the manuscript.
REFERENCIES Adler. J. (1976). 5%. rlr~er. 234 (4), 4&4i. Berg, H. (1. & Tedesco, P. M. (1975). I ‘roI‘. Nat. Acad. sci., 1 ..s.il 72. 3236-3239. Block, S. 31.. Segall, ,J. E. & Berg, H. (‘. (I 982). Cell, 31, 215-226. Block. S. M.. Segall, J. E. & Berg, H. C’. (1983). .I. Bactrriol. 154, 312-323. Boyd, A. & Simon, M. (1980). J. Bacterial. 143. 809-815. Boyd, A. & Simon, M. (1982). Annu. Rev. Physiol. 44, 501 -5li. Chelsky, D. C%Dahlquist, F. W. (1980). Proc. Sat. Acad. Ski.. l-X.A. 77. 2434-243X. Clarke, S. & Koshland, D. E. Jr (1979). .I. Biol. Chem. 254, 9695-9702. DeFranco, A. & Koshland, D. E. Jr (1980). Proc. Nut. Acad. Sci.. U.S.A. 77, 2429-2433. EngstrGm, P. & Hazelbauer, G. L. (1980). Cell, 20, 165-I 71. Goldbeter, A. & Koshland. D. E. Jr (1982). J. &!ol. Riol. 161. 39tiA18. Goy, M. F., Springer, M. S. & Adler, J. (1977). Proc. IVat. Acad. Xci.. I’.,Y.d 74. 4964~-496!) Uoy, M. F.! Springer, M. R. & Adler, ,J. (1979). CPU, 15, 1231-1240. Hayashi, H.. Koiwai, 0. & Kozuka, M. (1979). J. Biochem. (TO~YOJ, 85, 1213-1223. Hazelbauer, G. I,., Engstriim, P. & Harayama, S. (1981). J. Hacteriol. 145, 4339. Hedblom, M. L. & Adler, J. (1980). J. Hacteriol. 144, 1048-1060. Kehry, M. & Dahlquist, F. I). (1982). .J. Biol. Chem. 257, 10378-10388. Kihara, M. & Macnab, R. M. (1981). J. Bacterial. 145, 1209-1221. Kleene, S. J., Toews, M. L. & Adler, J. (1977). J. Biol. Chem. 252, 3214-3218. Koiwai, O., Minoshima, N. & Hayashi, H. (1980). J. Biochem. (Tokyo), 87. 1365.-1370. Kondoh, H.. Ball, (1. B. 85 Adler. J. (1979). Proc. Nut. Acad. Sci.. 1J.S.A. 76, 360-264. Kort, E. M., Goy, M. F., Larsen, S. H. &. Adler, ,J. (1975). Proc. ;VU~. Acad. ~Sci.. l’.S.A. 72.
3939-3943. Koshland, D. E. CJr (1980). Bacteriul Chew&axis us a Model Behavioral System, Ravrln Press, New York. Koshland, D. E. ,Jr (1981). Annu. Rev. Riochem. 50. 765.~782. Krikos, A., Mutoh, N., Boyd, A. & Simon, M. (1983). Cell, 33. 615-622. Macnab, R. M. (1978). CRC Grit. Rev. Biochem. 6, 291-341. Macnab, R. M. & Koshland, D. E. *Jr (1972). Proc. Nat. dead. Sci., I’.S.A. 69. 2509-2512. Maeda, K. & Imae, Y. (1979). Proc. Nat. Acad. Sci., C’4S.A. 76. 91-95. Maeda, K., Imae, Y.. Shioi, J.-T. & Oosawa, F. (1976). J. Bactrriol. 127, 103%1046. Mizuno, T. & Tmar. Y. (1984). J. Bacterial. in the press. Oosawa, K. & Tmae, Y. (1983). J. Bacterial. 154, 104-112. Parkinson, J. S. (1977). Annu. Rec. (:enet. 11. 397414. Parkinson, J. S. & Revello, P. T. (1978). Cell, 15, 1221L1230. Russo, A. F. & Koshland, D. E. Jr (1983). Sience, 220, 1015~1020. Silverman, M. & Simon, M. (1977). Proc. Nat. Acad. Sci., C’.S.A. 74, 3317.-3321. Slonczewski, J. L., Macnab, R. M., Alger. .J. R. & Castle, A. M. (1982). J. Bacterial. 152. 384-399. Springer, M. S., Goy, M. F. & Adler, ?J. (1977). Proc. ,Vat. Aca,d. Sci., I’.,S.A. 74. 3312-3316. Springer, M. S., Goy, M. F. & Adler. ,J. (1979). Nature (London), 280, 279-284. Springer, M. S., Zanolari, B. & Pierzchala. P. A. (1982). .J. Riol. Ch,pm. 257. 6861-6866. Springer, W. R. & Koshland. D. 6. .Jr (1977). Proc. Nat. Acad. Ski., /:.??.A. 74. ,533%537.
MODEL
FOR
BACTERIAL
CHEMORECEPTORS
X7
Spudich, J. L. & Koshland, D. E. Jr (1975). hoc. Nat. Ad. hi., U.S.A. 72, 710-713. Stock, J. R. & Koshland, D. E. Jr (1978). Proc. Nut. Acad. Sci., U.S.A. 75, 3659-3663. Stock, J. It. & Koshland, D. E. Jr (1981a). J. Biol. Chem. 256, 1082~10832. Stock, J. R. & Koshland, D. E. Jr (1981b). Cum. Topics Cell. Reg. 8, 505-517. Toews, M. I,., Goy, M. F., Springer, M. S. & Adler, J. (1979). Proc. Nut. Acud. Sci., 11.9.A. 76, 5544-5548.
Wang, E. & Koshland,
D. E. Jr (1980). Proc. Nut. Acud. Sci., U.S.A. 77, 7157-7161.
Edited by Q. A. Gilbert
Two-state
Model
for Bacterial
Chemoreceptor
Proteins
The Role of Multiple Methylation SHO ASAKURA AND HAJIME
HONDA
institute of Molecular Biology, Xchool of Science Nagoya University, Xagoya 464, Japan (Received
19 December 1983)
To help understand the bacterial chemotactic response of excitation and adaptation. we propose a simple two-state model for receptor proteins (methylaccepting chemotaxis proteins), in the light of evidence that they undergo multiple methylation in a preferred order. The model includes the following assumptions. (I) The receptor protein is in rapid equilibrium between two conformations, S and T, and the equilibrium shifts towards the T form as the number of methyl groups increases. (2) Attractants bind to the S form of the receptor. repellents bind to the T form, and both classes of ligand shift the S/T equilibrium according to the mass-action law. (3) The S form of the receptor accepts methyl groups one by one in a definite order. while the T form releases the methyl groups in the reverse order. Methylation and demethylation are slow reactions, and changes in the total number of methyl groups lag behind shifts in the S/T equilibrium. (4) The pattern of bacterial swimming at any moment, is determined by the partition of the receptor between the two conformations. with tumbling frequency being a monotonically increasing function of the total T fraction of the receptor. This model shows that, if the receptor satisfies two sets of relationships imposed on its equilibrium and kinetic constants, it can maintain the steady-state total T fraction essentially constant over a broad range of ligand concentration, enabling cells to adapt to large changes in chemical environment. A stepwise change in ligand concentration leads to a rapid change in the total T fraction (excitation). followed by a slow relaxation process (adaptation). Computer simulations have been made of the whole response process, employing a receptor with six methylation sites per molecule and assuming simple sets of parameters. The results are in general agreement with published data on receptor methylation, as well as with a variety of observations of bacterial chemoresponsr. Multiple methylation of the receptor proves to be necessary for the cells to respond sensitively to environmental changes.
1. Introduction lSacat,erial cells migrate through spatial concentration gradients of attractants and repellents. Typical attractants include serine, aspartate, maltose, rihosr and galactose: repellents include leucine, indole and ?ji2+. During chemot,actic migration, cells detect, via specific chemoreceptors, temporal changes in concentration of ligands, and they respond by altering their pattern of swimming t’hrough regulation of the sense of flagellar rotation. Counterclockwise rotation of
350
S. ASAKL~RA ASI) H. HONIIA
the flagella (as viewed from their distal end) produces smooth swimming. a11c1 clockwise rotation causes tumbling. The cells suppress tumbling for a periocl of’ time in response to attractant addition and repellent removal: conversely. the! enhance tumbling in response to attractant removal and repellent addition. Thrsr initial responses to environmental changes are followed by a period of adaptation in which the cells return to their pre-stimulus pattern of swimming. A remarkable feature of the phenomenon is that bacterial cells are able to adapt to a broad concentration range of chemical environments, but also to respond sensitively to small changes in them (for reviews. see Macnab. 1978: Springer r~f nl.. 1979: Koshland, 1980,198l; Boyd Ilr. Simon? 1982). In the bacterial chemoresponse of excitation and adapt’ation. a set of chemotaxis membrane-associated proteins, designated i.he methyl-accepting proteins or M(:Ps. play a central role (Kort’ c!t al., 1975; Springer et f~1.. 1979). Three MCI’s, all about 60,000 M,. have been ident,ified in Escherichia colt' (Springer et al., 1977; Silverman $ Simon, 1977; Kondoh rt nl., 1979; Koiwai et ul.. 1980). MCPT is necessary for response to serine. leucine and indole; MCPTl for response to aspartate, maltose and Xi2+: MCPIII for response to ribose and galactose. Methyl groups are transferred from S-adenosglmethionine to glutamyl residues in the MCI’s by a specific methyl-transferase. and the result’ing glutamylmethyl esters are hydrolyzed by a specific esterase (Kleene rt ul., 1977: Springer & Koshland, 1977; Coy et al.. 1977: Stock & Koshland. 197X). The MCPs art‘ multiply methylated (Boyd 8: Simon! 1980; Chelsky & Dahlquist. 1980: DeFranc*o & Koshland. 1980: Engstriim CyrHazelbauer. 1980): they have as many as six glutamyl residues per molecule that) are methylated in a preferred order (Kehry Q Dahlquist, 1982; Springer et al., 1982). The methyl-accepting glutamyl residue5 have been identified in the primary structures of MCPs (Krikos rt aZ.. 1983; Russo & Koshland, 1983). A single cell cont’ains more than 1000 molecules each of MCI’1 and MCPTT (Hayashi et ab., 1979; Stock & Koshland, 1981a). which account for, 900/:, of t,he t,otaI MCPs (Hazelbauer et nl.. 1981). L\:h en cells are in an isotropic chemical environment, the MCPs maintain a constant level of methylation, which in the presence of at’tractants (or repellents) is higher (or lower) than in their absence (Coy et al., 1977: Springer rt nl.. 1979). A wealth of evidence shows that MCPI and MCPTI are the serine and asparta,te receptors, respectively (Clarke & Koshland, 1979: Wang Br Koshland. 1980). and a mechanism for control rtf receptor methylation by ligand binding has been proposed by Stock B Koshland (1981a). Wit,h this experimenta,] background. we present here a simple two-si.atjcx model for the chemoreceptor protein. t’o throw light on its function at the initial step of sensory transduction in bact,eria. Goldbrter k Koshland (1982) have reportSed detailed analyses of a series of models for the receptor protein. Tttc present model is an altjernativr. 2. Results and Discussion (a) Model
The model is based on the following assumptions. (1) The receptor protein is in rapid equilibrium bet,ween two conformations. which are designated S and T because of their controlling effects on t.hc
MODEL
FOR
BACTERIAL
CHEMORECEPTORS
35 I
smooth/tumbly behavior of the motor organelles (see assumption (4)). The equilibrium shifts towards the T form as the number of methyl groups is increased. If the receptor protein molecule has a total of n sites that accept methyl groups one by one in a definite order, it partitions the methyl groups among (n+ 1) molecular species. The fraction of protein molecules present as the species having i methyl groups is denoted by Pi: $Opi
=
(1)
l.
Let Lp stand for the S/T equilibrium constant, in the absence of ligand, for the species having i methyl groups. Then, by assumption: L; > Ly >
> L,o .
(2)
(2) Attractants bind only to the S form of the receptor, and repellents bind only to the T form. Affinity between an attractant (or repellent) and the S (or T) form of t’he receptor is independent of the number of methyl groups, the dissociation constant being referred to as K, (or KR). Then, the mass-action law leads to the following relationships for any concentration of the attractant [A] or of the repellent [R] : (Al/K,
= [$,~A]/[&,]
= [S,.A]/[S,]
= . . . = lS;A]/[S,],
[RI/K, = [T,~RI/[R,I = [T,.RI/[T,I =
. = P’,~W[Tnl.
(3)
Normalized concentrations a = [AI/K, and /l = [RI/K, will be used hereafter in place of [A] and [RI. The sum of the [S,.A] and [Si] fractions is denoted by Si, and the sum of the [T,.R] and [Ti] fractions is designated Ti: Si + Ti = Pi.
(4)
Then, the Si/Ti ratios, referred to as Li terms, can be expressed as: ,1+ci Li = Si/Ti = L il+p
(i = 0, 1,.
) n).
Changes in ligand concentration lead to instantaneous changes in Li terms according to equation (5). Note t’hat, at any concentration of the attractant (or repellent), its overall affinity towards the receptor decreases rapidly with an increase (or decrease) in the number of methyl groups, since: L, > L, >
. > L,.
(6)
The sum of the Ti (or Si) values over all molecular species is denoted simply by T (or 8): T=
i Ti. i=O
Given the inter-species distribut’ion from:
s=
tsi
(T+s
= 1).
{Pi} at any moment, we can calculate T and S
T = t Pi/(1 +L,) and S = i P,L,/( 1 + Li), i=O
(‘i)
i=O
i=O
(8)
Xr’ and fractional
S. AS.-ZKI-RA AND H. HONDA receptor
occupancy ITA
zz
from :
s”-
1 +!x’
y R = 7’2 1+P
(3) The t)wo conformations of the receptor are recyyizrtl as being dist inc.t t)>the meth?il-t,ransferase and the esterase. The former enzyme catalyzes the transfer of methyl groups to each Si fraction (and not to ‘I‘, fractions). and the Iat trr enzyme cat.alyzes the removal of methyl groups from taac.1~Ti fraction (and IIOI from Si fract,ions), as depicted in the following scheme:
(IO]
where the sub-species of Si (or Ti) with and wit,hout bound l&and are assumed t)o be equally susceptible to methylation (or demcthylation). The methylation and demethylation are slow react.ions, and shifts in the inter-spec+s distribution lag behind changes in Li values. I)enot,ing the apparent rate c-onst,antb for met,h,ylation and demethylation at f,he ith step by CZ( and hi. then the rittcs equations governing the inter-species distribution are: I’” = -u,S,+h,7’,. Pi = -Ixi+l
Si+h;+,Ti+,
I’, = a, s,
, - h, T,
+Q\‘~
1~-h,T,.
ill)
These interdependent first-order differential equat)ions (aan h(b transformed into H independent &h-order differential equations. (4) The pattern of bacterial swimming at an!’ moment is tlet~erm~netf Ir)>. the partiCon of the receptor protein between thus t\vo (*onformations. a-ith 1he frequency of tumbling or the bias of flagellar rotation (the fraction of time spent rotating clockwise) being a monotonically increasing function of the total ‘I fraction, T. of the receptor. (To specify the details of t,his function. w-e nould hart* to elucidate the nature of the signal that is (Lontrolling the bias of flagellat motors.) This receptor model has elements in cbommon \vit,h thoscb proposed 1)~ lioshlanti (1981) and Stock & Koshland (1981h). which, however. have not been analyzrd mathematically. Tt should be mentioned that, the present model arose from caonsideration of the observed changes in switnming pattern of non-adaptive mutants. Cells missing the e&rave exhibit continuous tumbling. but they can hr made to swim smoothly by addition of attract)ants (Parkinson. 1977). (‘ells missing the transferase are usually smooth swimming hut can tumble frequrnt,l>. in the presence of repellents (Springer & Koshland. 1977: Parkinson & Revcllo. 1978: Coy et al.. 1979). These facts have led us t,o assume t’hat (1) the signal for
MODEL
FOR
BACTERIAL
353
CHEMORECEPTORR
tumbling is generated by the receptor regardless of the methylation level and the occupancy, and (2) attractants and repellents cause allosteric transitions in the receptor protein. According to this model, the changes in swimming pattern of a non-adaptive mutant with ligand concentration are governed simply by the massaction law, as the receptor is frozen either in the state that P, = 1 (for which it is assumed t,hat Lz K 1; see below) or in the state that P, = 1 (Li >>1). [Jnder steady-state conditions, equations (11) reduce to:
pi = ai/bi= T,/AS-~
(i = 1) 2,.
, n).
(12)
Combination of equations (4), (5) and (12) yields the steady-state’ Pi values as functions of ligand concentration. As the concentration of an attractant (or repellent) is increased, the steady-state inter-species distribution shifts towards the right-hand (or left-hand) side of the scheme of equation (lo), and these shifts are accompanied by a lowering of the overall afinity between the ligand and the receptor, as stated earlier. So, any ligand has its affinity for the receptor decreased in the steadyas the concentration is increased, leading to negative co-operativity stat’e binding. The co-operativity vanishes if methylation/demethylation is blocked. It will be seen shortly that, according to the model, the receptor can maintain its steady-state T fraction essentially constant over a broad range of attractant and repellent concentrations, provided the following two sets of conditions are satisfied simultaneously. (1) The S/T equilibria of the unmethylated and the fully methylated species heavily favor the S and T conformations, respectively: L~>>landL~<
(13) by the same ratio
pi = I* (constant) for all i. Under condition
of the rate
(14)
(2), equations (12) reduce to:
p = T,/S, = T,/S, = . . .= T,/S,-,
= (T-To)/@-S,),
(15)
where, under condition (l), both To and S, are virtually zero within a broad range of repellent and attractant concentrations. In this range, therefore, the receptor can maintain a constant steady-state T level at F = p/( 1 +p). This is an important property of this model that accounts for the capability of bacteria to adapt to large changes in ligand concentration (see below). Consider changes in Li, Pi and T brought about by a stepwise change in concentration from a( 1) to ~((2) at time t = 0. Steady-state values for a variable z before and after the change are referred to as z(1) and z(2), respectively, and the value immediately after the change is denoted by z(0). Then:
Li(0) = L,(2) and P,(O) = P,(l)
(i = 0, 1,.
) n)
(16’)
X-1
S. ASAKVRA
AND
H. HONI)A
and insertion of these relations in equation (8) yields: 7’(O) = i Pi(l)/{1
+I&)).
( 17)
i=O
If the steady-state Pi values have hrrn determined as functions c)f‘ t I~(, concentration of ligand. it is easy t,o calculate the T(O) valuc~ for any stepwisca stimulus from equation (17). The difference:
T(O)-T(2) = ~{Pi(l)-Pi(2)j:{l
+I&)).
!IH)
may be used as a measure of the receptor excitation evoked by a given stepwlst‘ stimulus. In order tha,t a stepwise stimulus should evoke strong excit#ation. therv should be a large difference in the steady-stat,e inter-species distribution hefort, and after the stimulus. Tn other words, t’he stead\--stat,e inter-species distributjion should strongly depend upon the concentration of ligand in order t)o make the receptor highly excit,able. This requirement is fillfilled hy virt,ue of multiple methylation (see below). The relaxatjion from 7’(O) to T(d). corresponding to adaptation, is governed by rate equations (1 I). which n-e solved hy comput’er IO simulate the process. The results are presented in the following section. (h) A nnlyxin Here we deal with t,he model numerically, employing the recept,or wit)h I/ = 6 and assuming simple set,s of parameters in acvorda,ncne with equations (1.3) and (14). The first example is: (‘asp A:
i
0
1
2
log I$
5.5
1.r,
3
4
;i
ti
I .o 0.5
0
-0.5
-4.5
p = 0.1: ui = I s- ’ and bi = IO s - ’ for all i values. Here. the values for I,: to IAt are distributed in simplicity. The use of the value 0.1 for p will parameter values, equations (5) and (1.5) lead partition of the receptor among the Lci and Ti attractant) and repellent. with .r = (1 +cr)j( 1 +/I): 0
i 6.
1p5r-2
7:- 1()-5,r-J
I 1OS- l l()-O.“,r-2
a geometric series merely fi)t be disrussed later. For thrsv to the following s&ad)--state species in an environment of
1
ti 1 t) 3 10 I oo.5.r 10 -()‘s,r2 IO 2,r3 IO- -.5 .x 4 1()-o’“/. ,()-l.s,r.2 lo- 3x2 ,f ’ 1
It is evident that, t,he steady-state statistical wrights of T, and 8, are virt nall~ zero over a broad range of /I and 2 values, although T, -+ 1 as /3 -+ x and S, + 1 as a -+ CD. From t’his sequence. it is easy t,o cal(ulatrl the inter-species distribution and the T value under steady&atr conditions. as functions of t,he concentration of liqand (Fig. 1). The calculated dependence of the inter-species distribution upon concentration is in general agreement with data obtained by gel electrophoretic resolution of the molecular species (Royd & Simon. 1980: (‘helsky & IIahlquist. 1980; DeFranco &, Koshland. 1980). With in the rang? of ligand concentration
MODEL
I’Olt
BA(‘TEKlAL
(‘HEMORE(‘EI’TORS
Log(lfa)
FIG. I. Steady-state properties of the receptor in case A, plotted against log (1 +G() and log (1 +/I). The fractions of the 7 possible molecular species. Pi values (thin continuous lines, the number on each c~arve indicating the number of methyl groups), the extent of met~hylation ,XiPJfi (dotted line). and the 7 Iwel (thick continuous line) are shown.
given in Figure 1, the T curve is approximately horizontal, with an average value T = I /I 1. More specifically, the T value remains constant, 0.0909+0.0009, over the range of IX < 103’5 and /I < 102’5. A deviation of T as small as 1y& would presumably not cause any observable change in the bias of flagellar motors, in which case cells can adapt to as large a range as a < 103.5 and fl < 102’5. This range extends further as Lg is decreased from 10-4.5 and Li is increased from 105,5. The fract’ional receptor occupancy (eyns (9)) at steady-state, within the adaptive range of ligand concentration in case A, can be approximated by: y,=--
10
c1
11 1+a’
1 yR=llI+p.
B
Therefore, the steady-state fractional occupancy appears to increase in a hyperbolic manner with ligand concentration, approaching a limiting value smaller t’han unity, as if a part of the receptor were sequestered. This behavior, is due to t’he negatively co-operat’ive which may be termed pseudo-saturation, ligand binding mediated by methylation/demethylation. Measurements of t.hr pseudo-saturation curves yield the dissociation constants K, and K,. It’ is t.o he expected that, when this procedure is used to compare a wild-type strain and a non-adaptive mutant strain lacking the transferase (or esterase), the same K, (or KR) value will be obtained from them. The rate of methanol production, which was first measured by Toews cut rcl. (1979), corresponds to equation (20):
J = xbi7’i.
S. ASAKL’ILA
AND
0 Loqiltp)
H.
HO91)A
I 2
1 I
I 3
L.3qt1taI
Fro. 2. Changes in I[’ upon applying stepwiar changes in ligand concentration continuous and broken lines represent the T(O) values when ligand concentration decreased to zero, respectively. Thick continuous line: the steady-state T level.
in case A. The thm is increased from and
In case A, where we assume bi = 6 (constant) for all i values. tjhe steady-state J are constant, at the value values over the adaptive range of ligand concentration bT. This is consistent with the observation by Toews rt al. (I 979) that the steady state rate of methanol production in the presence of high concentrations of attractants was virtually equal to that in their absence. However, it may be an over-simplification to allot the same value to t’he b, terms, alt’hough ai/bi ratios at all methylation steps must be constant. Suppose that h, is larger t’han t,he other bi t,erms, while ai/bi = 0.1 for all i values as in case A. Then, under steady-state conditions, J will have a maximum value at log (I + 2) = 0.75 (x = 4%), because T, is at’ its maximum at t’his concentration. Tncidentally, the steady-state values for T, and P, have maxima at CLvalues close to each other (Fig. 1). Transient changes in T upon applying stepwise changes in ligand concentration are shown in Figure 2. Increases in ~1and decreases in /I lead to negative values for T(O)-T(2) (corresponding to a decrease in tumbling frequency). decreases in r and increases in /I lead to positive values (enhanced tumbling frequency), and large concentration differences give rise to large values for IT(O)-T(2)/. These results suggest that the model may work well to describe excitation. Computer simulations have been made of the T(O) -+ T(2) process in case A, where ai= Is-’ and hi= 19s -I for all i values. The time-scale should be considered arbitrary. Although the assumed set of rate constants is not necessarily realistic, the results make clear the general transient properties of the receptor. It is seen from Figure 3 that the excitation evoked by addition of a high concentration of an attractant (or repellent) relaxes considerably more slowly (or rapidly) than that evoked by its removal, in agreement with the well-known observations by Macnab & Koshland (1972) and by Berg c1:Tedesco (1975). The
0.
I-
MODEL
FOR
HA(:TERlAL
-
Attractant addition
CHEMORECEPTOKS
Xii
Repellent
T
()Removal
T
Add’hcn
0.I 5-
(I-
l 0
I
I
I
I
4
2
FIG 3. Relaxation processes of 7’ in case A. Ligand addition: stepwise increase in a or /I from 0 to 99 rep at f = 0. and removal: decwasr from 99 to 0. Time was divided into 042 s intervals for soloing (I 1) hy
computer.
relaxat~ion may he characterized in terms of the half-time t1,2 (the time required for (T-T(2)}/{?‘(0)Y’(2)) to reach the value l/S). The tIiz values calculated for various kinds and amplitudes of stepwise stimuli are shown in Figure 4. The four curves appear to approach limiting levels with increases in stimulus amplitude. r-
I-
J; 19
i
--\
\
I
\
‘\ c)-
I
I
2
I
---f-k-
Log (I tp,
FIG. 1. The I,,, values ralculated linrs) and for decrrases tn
(continuous
for zero
‘---- _ LogCltCY)
stepwise (hrokrn
increases lines)
in case
in lipand A.
concentration
hm
7rw.1
3%
S. ASAKI‘ltA
AND
H.
HOSIIA
The ratio between the limiting levels for addition and removal of an attractant IS about 12. This ratio increases further as the parameter p is decreased from 0.1. times for addiGon of’ high Berg &, Tedesco (1975) showed that, transition c:oncent.rations of an attractant were 100 times as long as those for its remova,l. III view of this. the true ,Dvalue may be smaller than 0.1. In F’igurcs 4, the four (*urvt+ start from the same height, of the ordinate, t,/, = (M4 seconds. This implies that excitations evoked by very sma.11amplitude stimuli of attracat’ant and repellent relax at a finit,e rate (see below). The magnitude of the response of a receptor caan be c~xprrssc~tlin ttarrns of’ A = i (7’-7’(2)]dt. 0
(“I)
which may be positive or negative depending upon the nature of a given stimulus. Figure 5 shows the A values calculated for various kinds and amplitudes of stimuli in ease A. The two sets of A curves in the first and fourth quadrants and those in the second and third quadrants are approximately mirror images. respectively. Let A,, stand for response to a stepwise stimulus, c1(1) -+ ~(2) or /?(I) -+ p(2). and let 6 be:
0.6
I
FIG. 5. Thr integrated response (A) values wlculated for case A. The 3 wntinuous rurvc~ in thr 4th quadrant (lower right) represent the A values for stepwise increases in 1 from 0. 9 and 99 to thts indicat,ed values. The curves in the first quadrant (upper right) represent the values for decreases in z from the indicated values to 0, 9 and 99. The 2 sets of curves in the second (upper left) and 3rd (lower left) quadrants, obtained for repellent stimuli. have similar meanings Integration of eqn (21) was function of st,opped when ld!J’/dtl became less than 2 x IO- ’ s -I The dotted line indicates a hyperbolic a with the half-maximum at c( = I, which represents the steady-state receptor occupanyv in t,hr adaptive range of &and concentration.
MODEL
FOR
BACTERIAL
CHEMORECEPTORR
350
Fro. 6. Changes in 7’ brought about by addition of an attractant (CL= 0 + 99) at t = 0, followed by its removal (99 + 0) at t = 0.5, 1.0 and 1.5 s, in case A. In each response curve, the area of the positivr part is larger than that of the negative part; however. the difference is smaller than 5O;,.
Then, 6 < 0.01 in the cc-region, and 6 < 0.07 in the P-region. Furthermore, the A curves in each quadrant can be superimposed closely upon each other by translation along the ordinate. Let 6’ stand for: (22b) Then, 6’ < 0.002 for increases in a, and 6’ < O-02 for decreases in b. Thus, A values and decrements in ligand concentration are for consecutive increments approximately additive. Figure 6 shows responses to attractant pulses of various widths. Each T curve consists of negative and positive parts (as measured from the steady-state level), the areas of the two parts being approximately equal (6 < 0.05). These and many other examples indicate that responses are approximately additive regardless of the intervals of time between successive st,imuli. The additivity of receptor responses reminds us of the additivity of bacterial responses as observed by measurements of recovery times (Spudich &, Koshland, 1975) and of transition times (Berg & Tedesco, 1975). However, before discussing the relationship bet’ween the experimental and theoretical results: we have to specify the functional T-dependence of the frequency of tumbling or the bias of flagellar motors. It is seen from Figure 5 that response to attractant, addition (the outermost continuous curve in the fourth quadrant) is not proportional to the steady-state receptor occupancy (the dotted line). We conclude therefore that, in the present receptor model, the additivity of responses is not based upon the proportionality between responses and changes in occupancy. Goldbeter & Koshland (1982) h ave also argued that the property of
additivity can lie dissociated from the l~ro~~ot~tionulit~~ l)rtween responses itrlti changes in occupancy. For very small stepwise stimuli, receptor responses are proportional to stimulus amplitude. and arc additive (Fig. 7). Lye obtain a value fbr the ~)roportionalit.!~ constant dlAl/dcr or dA/dj of 0.14 seconds, which tnay be regarded as the resl~msc sensitivity of the recept,or (case .A) at low concentration of ligand. Transient changes in 7’ upon applying very small stepwise stimuli are also proportional IO stimulus amplitudes (Fig. 7). This means that attractant and repellent stimuli of’ very small amplitudes evoke excitations that relax at a finite rate. The same conclusion has been drawn from the calculat’ion of t 1,2 (Fig. 1). These results are in accordance with some aspe& of data on impulse responses of wild-type ~~11s. reported by Block ef nl. (1982), although the existence of a stimulus threshold for evoking excitation cannot be explained by the present model. LVe should mention at t’his point that a simplitied version of t,hc receptor \rith II = 1 (L,O >> 1 and I,: << 1) can maintain a constant stead!--state T level over a broad range of l&and concentration. When, for example. /$ = 10s’“. I,? = 10 m4’i and p1 = 0.1, the steady-state T value is kept constant (0.0909 f 04009) ovet x < 103’5 and p < 102’5. as in case A. In this range of ligand concentratioii. however. the steady-state inter-species distribution remains virtually caonstant changes in 7 (PO = 0.9009 - 0.9092 and P1 = 04991 - 0.0908). and transient upon applying large st~epwise stimuli are negligible as compared to the stradystate level. That, is. the YI = 1 receptor (*annot work to excite the cells. \l’t. calculated A values for the receptor with YI = 3. assuming 1,: = 105’5. Ly = 10“5. At IOM z,; = 10-“‘5. Li = 10-4’5. )I CL-= 1 b‘i-l and hi = 10si - ’ for all i valnrs.
MODELFORBACTERIALCHEMORECEPTORR
XI
concentration of ligand, the response sensitivity of this receptor is 4.0-fold smaller than that in case A. It turns out from these comparisons that, in receptor A, the five molecular species, P, to P,, are co-operating to increase the response sensitivity within a range of (x < 103” and p < lo*“. where the steady-state T level is kept constant by the existence of the two extreme species. In conclusion. multiple methylation of the receptor plays a dual role: it allows the cells to adapt to large changes in ligand concentration, and it also increases their excitation sensitivity to small changes. For the second example of the n = 6 receptor, we assume the following set of parameters: Case B:
i log
0
1
2
45
3
6
L; 5.5 1.0 0.75 0.5 0.25 0 -4.5
a,=1s-’
and bi = 10 s-l
for all i values,
more closely than in case A. It is where the values for Ly to Lt are distributed seen from Figure 8(a) that the steady-state Pi curves in case B are located more closely than in A. The steady-state T levels in the two cases coincide over a range of CI < lo3 and p < 103. The A curves calculated for the two cases are compared in Figure 8(b). At low concentration of ligand, the response sensitivity of receptor H is 1.7.fold higher than that of A, while the sensitivity of receptor B decreases wit’h ligand concentration more rapidly than that of $. So, the former receptor is not) necessarily superior to the latter. Also, in case B, responses are approximately additive, although the response to attractant addition is not proportional to the steady-state occupancy (Fig. 8(b)). The third example is: Case C:
i log L;
0
1
2
3
5.5 1.25 1.0 0.75
ai = 1 s-l
and bi = 10 s-l
4
5
6
- 1.25 - 1.5 -4.5 for all i values,
where the L:/Lz ratio is assumed to be large. This non-uniform distribution of Li values gives rise to large P, values at intermediate concentrations of an attractant under steady-state conditions (Fig. 9(a)), leading to a biphasic A curve for its addition (Fig. 9(b)). In this case also, the steady-state T level remains constant over a < 103” and /? < lo*“, where the steady-state occupancy is simply hyperbolic. It is well-known that bacteria respond to increasing concentrations of serine in a biphasic manner (Springer et al., 1977; Hedblom & Adler, 1980). To interpret this phenomenon, it is usually assumed that the serine receptor has two sites with different dissociation constants. However, the above example suggests t’hat even a single binding site may, given the appropriate characteristics of t’he multiple methylation system, produce the biphasic response. The model can be applied to simulate receptor responses during continuous changes in ligand concentration. Figure 10(a) shows the response of receptor A to a sinusoidal change in attractant concentration of the form: a = 0.5{1-cos(27$)},
where f = 517 s-l.
(23)
T
o.!
Log iita)
Log (I cp, (0)
The calculated 7’ c~~rve is periodic and becomes stationary a fiw periods after the initiation of stimulus. The stationary waves rousist of tlw psitivr and negativr~ half-periods (as measured from the steady-&d Iwrl) that have the same area hut different amplitudes. Figure IO(b) is a tloublr logarithmic* plot of t,ht, normalized
MODEL
FOR
BACTERIAL
CHEMORECEPTORS
I*C )r
------A T
o-c:
\
Log(I+P)
Log (Ita)
Log cutp, (b)
FIG. 9. (a) Steady-state properties of the receptor in case C. For an explanation, see Fig. 1. (b) The IAJ curves calculated for attractant addition and repellent removal in case C (continuous lines). The broken lines represent the [A\ curves calculated when Li = 1Oo’5 and Lt = 10°‘25, with the other parameters being unchanged from case C.
mean stationary amplitude of the response as a function of frequency f (Bode plot). The results resemble the amplitude gain of a first-order high-pass filter. Block et al. (1983) have measured bacterial responses to exponentiated sine waves of an attractant at the low frequency side, and have found that the response was a first-order high-pass process. The combination of the experimental and theoretical results implies that the rate of bacterial response to a gradual change in ligand concentration is determined by receptor methylation.
304
S. ASAKIJKA
0
AND
H. HOSI)A
2
4 f, (0)
1
I-
0.01
I 0.01
I 0.1 f lb)
FIG. 10. (a) Change in T of receptor A in response to a sinusoidal change in at,tractant concentration given by eqn (23). The stimulus starts at t = 0. Abscissa: time multiplied by frequencyf = 5/7 SC’. The response wave becomes stationary a few periods after the initiation of stimulus. The mean amplitude of the stationary wave tends to 0.052 at high frequency. which coincides with the mean of the values for T(O)-Z’(2) for 2 independent stepwise changes, 0.5 + 0 and 0.5 + 1. (b) A double logarithmic plot of the normalized mean stationary amplitude as a function of f. The continuous line indicates the amplitude gain of a first-order high-pass filter.
Bacterial cells undergo thermotaxis (Adler, 1976; Maeda et al.. absence of ligand, they suppress (or enhance) tumbling frequency time in response to temperature increase (or decrease), and adapt. evidence shows that the thermal stimulus is detected mainly receptor and marginally by the aspartate receptor (Maeda & Imae, & Tmae, 1984). On these grounds, we assume that the equilibrium
1976). In the for a period of Experimental by the serine 1979; Mizuno constants, Lf’.
MODEL
FOR
I o
2
BACTERIAL
I I Log (I +p,
CHEMORECEPTORS
0
1 I
I 2
Log(lfa)
FIG. 11. Thermal excitation of a receptor that has the 2 sets of parameters, of cases A and R, at a given higher and lower temperature, when it is subjected to temperature jumps between the 2 values at various ligand concentrations. The thin continuous and broken lines indicate the T values Thick continuous line: the immediately after temperature increases and decreases, respectively. steady-state 2’ level. The 2 transient T curves intersect the steady-state curve at different points.
for each of these receptors depend upon temperature, while the parameter p is independent of temperature. If this assumption is correct, a temperature jump in the presence or absence of ligand would bring about instantaneous changes in L, (eqns (5)) for a given receptor, leading to a transient change in T. If two sets of Lp values of the receptor at two different temperatures are given, transient changes in T upon applying temperature jumps between the two values at various ligand concentrations can be calculated from equation (17). Figure 11 shows the results obtained for a receptor that has the two sets of parameters, of cases A and B, at a given higher and lower temperature, respectively. The thermal excitation is inverted as attractant concentration increases. This result is in agreement with the finding by Mizuno & Imae (1984) that mutant cells lacking the serine receptor protein exhibited inverted thermoresponse when they had become adapted to intermediate concentrations of aspartate. Therefore, it is likely that the distribution of Lp values in the aspartate receptor becomes broader with a rise in temperature. The present model may apply also to bacterial responses to weak acids (Kihara & Macnab, 1981; Slonczewski et al., 1982) and to glycerol and ethylene glycol (Oosawa & Imae, 1983). The nature of the signal controlling the bias of flagellar motors remains obscure. The elucidation of its nature is awaited for developing the model by determination of the functional dependence of the bias of motors upon T. It is probable that the serine, aspartate and ribose receptors are controlling motors by means of a common signal (such as a specific small molecule or ion). If so, the bias of motors may possibly depend upon “T” with respect to the total receptor, rather than for individuals, and the dependence around the steady-state in the absence of ligand is supposed to be very sharp, because ribose addition excites
XC
S. ASAKl:KA
AXI)
H. HONDA
cells to the full extent, whereas the ribose receptor (MCIPIII) amounts to onI> 10% of the total. From t,hese considerations, it is to be expected that t,herr is ii second mechanism for signal amplification in the bacterial chemoresponse. WI: thank Drs \‘. Imae, S. Kobayashi, H. Hayashi and I’rofcssor 11’. Oosawa 01’ this Institute for stimulating discussions. Thanks are also due to Prof&saor R. M. Macnab (Yale University) for critically reading the manuscript.
REFERENCIES Adler. J. (1976). 5%. rlr~er. 234 (4), 4&4i. Berg, H. (1. & Tedesco, P. M. (1975). I ‘roI‘. Nat. Acad. sci., 1 ..s.il 72. 3236-3239. Block, S. 31.. Segall, ,J. E. & Berg, H. (‘. (I 982). Cell, 31, 215-226. Block. S. M.. Segall, J. E. & Berg, H. C’. (1983). .I. Bactrriol. 154, 312-323. Boyd, A. & Simon, M. (1980). J. Bacterial. 143. 809-815. Boyd, A. & Simon, M. (1982). Annu. Rev. Physiol. 44, 501 -5li. Chelsky, D. C%Dahlquist, F. W. (1980). Proc. Sat. Acad. Ski.. l-X.A. 77. 2434-243X. Clarke, S. & Koshland, D. E. Jr (1979). .I. Biol. Chem. 254, 9695-9702. DeFranco, A. & Koshland, D. E. Jr (1980). Proc. Nut. Acad. Sci.. U.S.A. 77, 2429-2433. EngstrGm, P. & Hazelbauer, G. L. (1980). Cell, 20, 165-I 71. Goldbeter, A. & Koshland. D. E. Jr (1982). J. &!ol. Riol. 161. 39tiA18. Goy, M. F., Springer, M. S. & Adler, J. (1977). Proc. IVat. Acad. Xci.. I’.,Y.d 74. 4964~-496!) Uoy, M. F.! Springer, M. R. & Adler, ,J. (1979). CPU, 15, 1231-1240. Hayashi, H.. Koiwai, 0. & Kozuka, M. (1979). J. Biochem. (TO~YOJ, 85, 1213-1223. Hazelbauer, G. I,., Engstriim, P. & Harayama, S. (1981). J. Hacteriol. 145, 4339. Hedblom, M. L. & Adler, J. (1980). J. Hacteriol. 144, 1048-1060. Kehry, M. & Dahlquist, F. I). (1982). .J. Biol. Chem. 257, 10378-10388. Kihara, M. & Macnab, R. M. (1981). J. Bacterial. 145, 1209-1221. Kleene, S. J., Toews, M. L. & Adler, J. (1977). J. Biol. Chem. 252, 3214-3218. Koiwai, O., Minoshima, N. & Hayashi, H. (1980). J. Biochem. (Tokyo), 87. 1365.-1370. Kondoh, H.. Ball, (1. B. 85 Adler. J. (1979). Proc. Nut. Acad. Sci.. 1J.S.A. 76, 360-264. Kort, E. M., Goy, M. F., Larsen, S. H. &. Adler, ,J. (1975). Proc. ;VU~. Acad. ~Sci.. l’.S.A. 72.
3939-3943. Koshland, D. E. CJr (1980). Bacteriul Chew&axis us a Model Behavioral System, Ravrln Press, New York. Koshland, D. E. ,Jr (1981). Annu. Rev. Riochem. 50. 765.~782. Krikos, A., Mutoh, N., Boyd, A. & Simon, M. (1983). Cell, 33. 615-622. Macnab, R. M. (1978). CRC Grit. Rev. Biochem. 6, 291-341. Macnab, R. M. & Koshland, D. E. *Jr (1972). Proc. Nat. dead. Sci., I’.S.A. 69. 2509-2512. Maeda, K. & Imae, Y. (1979). Proc. Nat. Acad. Sci., C’4S.A. 76. 91-95. Maeda, K., Imae, Y.. Shioi, J.-T. & Oosawa, F. (1976). J. Bactrriol. 127, 103%1046. Mizuno, T. & Tmar. Y. (1984). J. Bacterial. in the press. Oosawa, K. & Tmae, Y. (1983). J. Bacterial. 154, 104-112. Parkinson, J. S. (1977). Annu. Rec. (:enet. 11. 397414. Parkinson, J. S. & Revello, P. T. (1978). Cell, 15, 1221L1230. Russo, A. F. & Koshland, D. E. Jr (1983). Sience, 220, 1015~1020. Silverman, M. & Simon, M. (1977). Proc. Nat. Acad. Sci., C’.S.A. 74, 3317.-3321. Slonczewski, J. L., Macnab, R. M., Alger. .J. R. & Castle, A. M. (1982). J. Bacterial. 152. 384-399. Springer, M. S., Goy, M. F. & Adler, ?J. (1977). Proc. ,Vat. Aca,d. Sci., I’.,S.A. 74. 3312-3316. Springer, M. S., Goy, M. F. & Adler. ,J. (1979). Nature (London), 280, 279-284. Springer, M. S., Zanolari, B. & Pierzchala. P. A. (1982). .J. Riol. Ch,pm. 257. 6861-6866. Springer, W. R. & Koshland. D. 6. .Jr (1977). Proc. Nat. Acad. Ski., /:.??.A. 74. ,533%537.
MODEL
FOR
BACTERIAL
CHEMORECEPTORS
X7
Spudich, J. L. & Koshland, D. E. Jr (1975). hoc. Nat. Ad. hi., U.S.A. 72, 710-713. Stock, J. R. & Koshland, D. E. Jr (1978). Proc. Nut. Acad. Sci., U.S.A. 75, 3659-3663. Stock, J. It. & Koshland, D. E. Jr (1981a). J. Biol. Chem. 256, 1082~10832. Stock, J. R. & Koshland, D. E. Jr (1981b). Cum. Topics Cell. Reg. 8, 505-517. Toews, M. I,., Goy, M. F., Springer, M. S. & Adler, J. (1979). Proc. Nut. Acud. Sci., 11.9.A. 76, 5544-5548.
Wang, E. & Koshland,
D. E. Jr (1980). Proc. Nut. Acud. Sci., U.S.A. 77, 7157-7161.
Edited by Q. A. Gilbert