In this lecture we will discuss and contrast two models that model bacterial chemotaxis.

Transcription

1 VIII Modeling Escherichia coli chemotaxis In this lecture we will discuss and contrast two models that model acterial chemotaxis. References: 1. P. A. Siro, J. S. Parinson, and H. G. Othmer. A model of excitation and adatation in acterial chemotaxis. PNAS 94, (1997). 2. N. Barai and S. Leiler. Roustness in simle iochemical networs. Nature 387, (1997). In Siro s model the Tar recetor always forms a comlex with CheA and CheW. CheW functions as an adater (scaffolding) rotein and has no enzymatic function. The comlex has two hoshorylation states (due to CheA), three methylation states, and ligand ound or unound state. These 12 different states are summarized in Fig. 2 of Siro s aer. The ligand (un)inding reactions are the fast reactions (millisecond) whereas the methylation reaction are slow (minutes). The hoshorylation reactions san the intermediate time scales. Below we will exlore Siro s model and try to inoint why this model has to e fine tuned in order to reroduce erfect adatation. This is in contrast to Barai s model (see elow) that does not need fine tuning to otain erfect adatation. First, let us assume that we only have to consider two methylation states (2 and 3 methyl grous). Including more methylation states does not fundamentally change the roerties of the model. We can always assume that we oerate at low concentrations of external ligand so that only the low methylation states will e relevant. Rememer that the numer of methylated sites increases with increasing ligand concentration. With this assumtion the 12 different recetor states reduce to 8 states /7.81J/8.591J Systems Biology A. van Oudenaarden MIT Octoer 2009

2 Secondly, the time scale of ligand inding and uninding is almost three orders of magnitude faster than the hoshorylation and methylation times. We can therefore treat the ligand (un)inding reactions as equiliria. This reduces the ossile recetor states to 4 (Fig. 16, Matla code 5). T 2 LT 2 eff4 eff3 T 3 LT 3 eff1 t t T eff4 2 T 3 LT 2 LT 3 eff3 Figure 16. Reduced version of Siro s model. The fraction of recetors that are ound to a ligand f can e written as (analogous to [II.12], n = 1): f KL = [VIII.1] 1+ K L where L is the ligand concentration and K is the association constant for ligand inding: K = = = = 10 M [VIII.2] The effective rates are weighted averages of the rates given y Siro: /7.81J/8.591J Systems Biology A. van Oudenaarden MIT Octoer 2009

3 eff1 eff3 = (1 f ) + f 8 = (1 f ) + f = 9-1 (1 f ) f K L = 1+ K L K L = 1+ K L KL = 1+ K L [VIII.3] The rates on the right hand side are the rates defined in Siro s aer. The effective methylation rate can not e written down y a single effective rate constant as methylation is assumed to oey Michaelis Menten inetics. The methylation rates of the non hoshorylated and hoshorylated recetors are, resectively: v r = K r R v = K R (1 f )[2] v + + (1 f )[2] K max1 (1 f )[2] v + + (1 f )[2 ] K max1 R f[2] + f [2] max3 max3 R f [2 + f [2 ] ] [VIII.4] where [2] and [2 ] are the total concentrations of non hoshorylated and hoshorylated recetors with two methylation sites. The maximum turnover rates are V max1 = 1c R and V max3 = 3c R, where R is the total amount of CheR. K R is the Michaelis constant for recetor CheR inding (1.7 μm). Note that the hoshotransfer rate is indeendent of L. t = (Y Y ) + (B B ) [VIII.5] y o o What is needed for erfect adatation? Can we write a general relation that tells us how to fine tune the rate constants? Suose the methylation rates [VIII.4] would oey ordinary first order inetics, in this case we can write simle ratios etween the different recetor states: [2] [3 ] [3 ] eff1 [3] eff4 =, =, = = [VIII.6] [2] [3] [2] [2 ] t t eff /7.81J/8.591J Systems Biology A. van Oudenaarden MIT Octoer 2009

4 This system is over determined (4 unnowns, 5 equations) since the total amount of recetor is fixed. In other words no steady state solution exists. By assuming Michaelis Menten inetics [VIII.4] you can introduce one additional variale that solves this issue. Let s go ac to the erfect adatation. Perfect adatation means that in steady state the numer of hoshorylated recetors is indeendent of the ligand concentration: the effective hoshorylation rate is indeendent of ligand concentration. On long time scale the networ will equilirate having a fraction (1 α) in state [2] and a fraction α in state [3]. The net hoshorylation rate will then e: = (1 α) + α [VIII.7] hos eff1 To otain erfect adatation α should e: hos eff1 hos(1+ KBL) 8 11K BL α = = [VIII.8] ( ) + ( )K L eff1 9 8 The main oint is that it is very difficult to otain erfect adatation in this model. It wors for a very secific set of constants, ut small variations from this set will lead to non erfect adatation B Barai s model uses a similar aroach ut differs in a sutle way y maing crucial different assumtions. The main difference is that CheB only demethylates hoshorylated ( active ) recetors. As in Siro s model, Barai s model can reduced to four states (Fig. 17). T 2 LT 2 eff3 eff4 T 3 LT 3 eff1 t t T eff3 2 T 3 LT 2 LT 3 Figure 17. Stried down version of Barai s model /7.81J/8.591J Systems Biology A. van Oudenaarden MIT Octoer 2009

5 A second imortant assumtion is that the methylation rates oerate at saturation since [CheR] is much smaller than the concentration of recetors. This means that methylation rate is constant and is indeendent of [2] and [2 ]. The final crucial assumtion is that demethylation is indeendent of ligand inding. This leads to the reduce scheme deicted in Fig. 18. r in eff4 T 3 LT 3 t r in T 3 LT 3 Figure 18. Even further stried down version of Barai s model. Where r in is the saturated methylation rate that is indeendent of [2] and [2 ]. The inetic equations for these reactions are: d[3] = r dt d[3] = rin dt in + t eff4 [3 ] - [3 ] t [3 ] + [3] [3] [VIII.9] The total amount of recetor evolves according to: d[3 d[3 ] T ] d[3] = + = 2rin - eff4[3] [VIII.10] dt dt dt In steady state this means that the concentration of [3 ] is: 2r in [3 ] = [VIII.11] eff4 indeendent of the roerties of the hoshorylation reaction and external ligand concentration. This system will therefore oey erfect adatation, for any change in ligand concentration /7.81J/8.591J Systems Biology A. van Oudenaarden MIT Octoer 2009

6 Figure 19. Uer anel: erfect adatation is oserved for certain values of the net hoshorylation rate if 11 > 9. Perfect adatation is not oserved if 11 < 9 (lower anel). See equation [VIII.3] for definition of rate constants /7.81J/8.591J Systems Biology A. van Oudenaarden MIT Octoer 2009

7 Matla code 5: Siro model % filename siro.m clear; close; To=8e-6; Yo=20e-6; Bo=1.7e-6; otions = odeset('reltol',1e-9,'astol',[1e-9 1e-9 1e-9 1e-9 1e-9]); [t y]=ode23('sirofunc',[0 80],[4e-6 4e-6 0e-6 0.5e-6 10e-6],otions); Ptot=To-y(:,1)-y(:,2); B=y(:,4); Y=y(:,5); metlevel=1-(y(:,1)+y(:,3))/to; hoslevel=1-(y(:,1)+y(:,2))/to; sulot(2,2,1) lot(t,hoslevel,'x'); axis([ ]); title('phoshorylation Level'); sulot(2,2,2) lot(t,metlevel,'rx'); title('methylation Level'); axis([ ]); sulot(2,2,3) lot(t,b/bo,'gx'); axis([ ]); title('b/btot'); sulot(2,2,4) lot(t,y/yo,'yx'); axis([ ]); title('y/ytot'); /7.81J/8.591J Systems Biology A. van Oudenaarden MIT Octoer 2009

8 %filename sirofunc.m function dydt = f(t,y,flag) % constants from Tale 3 (Siro et al.) 1c=0.17; % 1/s 3c=30*1c; % 1/s ratio11a=1.7e-6; ratio3c3a=1.7e-6; _1=4e5; _3=_1; 8=15; % 1/s 9=3*8; % 1/s 11=0; % 1/s %12=1.1*8; % 1/s 12=30; =8e5; y=3e7; _=0.35; % 1/s _y=5e5; Kind=1e6; % 1/M Yo=20e-6; Bo=1.7e-6; To=8e-6; Ro=0.3e-6; Zo=40e-6; % [T2]+[LT2] = y(1) % [T3]+[LT3] = y(2) % [T2]+[LT2] = y(3) % [B] = y(4) % [Y] = y(5) cligand=1e-6; if t>20 cligand=1e-3; end; if t>50 cligand=1e-6; end; Vmaxunound=1c*Ro; % maximum turnover rate (MM inetics) for unound recetors Vmaxound=3c*Ro; % maximum turnover rate (MM inetics) for ound recetors KR=ratio11a; ichaelis constant f=kind*cligand/(1+kind*cligand); % fraction recetors ound to ligand fu=1-f; % fraction recetors not ound to ligand t=y*(yo-y(5))+*(bo-y(4)); ydot1=(-8*fu-11*f)*y(1)+t*y(3)+(_1*fu+_3*f)*y(2)*y(4)- Vmaxunound*y(1)*fu/(KR+y(1)*fu)-Vmaxound*y(1)*f/(KR+y(1)*f); ydot2=(-9*fu-12*f)*y(2)+t*(to-y(1)-y(2)-y(3))- (_1*fu+_3*f)*y(2)*y(4)+Vmaxunound*y(1)*fu/(KR+y(1)*fu)+Vmaxound*y(1)*f /(KR+y(1)*f); ydot3=(8*fu+11*f)*y(1)-t*y(3)+(_1*fu+_3*f)*(to-y(1)-y(2)- y(3))*y(4)-vmaxunound*y(3)*fu/(kr+y(3)*fu)+vmaxound*y(3)*f/(kr+y(3)*f); ydot4=*(to-y(1)-y(2))*(bo-y(4))-_*y(4); ydot5=y*(to-y(1)-y(2))*(yo-y(5))-_y*y(5)*zo; dydt=[ydot1; ydot2; ydot3; ydot4; ydot5]; /7.81J/8.591J Systems Biology A. van Oudenaarden MIT Octoer 2009