near-perfect adaptation in bacterial chemotaxis

04/21/23 Yang Yang, Candidacy Seminar 1

Near-Perfect Adaptation in Bacterial Chemotaxis

Yang Yang and Sima Setayeshgar

Department of Physics

Indiana University, Bloomington, IN

E. coli and Bacteria Chemotaxis


http://www.rowland.harvard.edu/labs/bacteria/index_movies.html

Increasing attractants or Decreasing repellents

Chemotaxis Signal Transduction Network in E. coli

04/21/23 Yang Yang, Candidacy Seminar

5

Histidine kinase Methylesterase

Couples CheA to MCPs Response regulator

Methyltransferase Dephosphorylates CheY-P

CheB

CheW

CheZ

CheR

CheY

Signal Transduction

Pathway

Motor Response

[CheY-P]

Stimulus

Flagellar Bundling

Motion

Run Tumble

Robust Perfect Adaptation

04/21/23 Yang Yang, Candidacy Seminar

Fast response Slow adaptation

From Sourjik et al., PNAS (2002).

FRET signal [CheY-P]

From Alon et al., Nature (1999).

CheR fold expressionAd

apta

tio

n

Pre

ciso

n

Steady state [CheY-P] / running bias independent of value constant external stimulus (adaptation)

Precision of adaptation insensitive to changes in network parameters (robustness)

6

This Work: Outline


New computational scheme for determining conditions and numerical ranges for parameters allowing robust (near-)perfect adaptation in the E. coli chemotaxis network

Comparison of results with previous works

Extension to other modified chemotaxis networks, with additional protein components

Conclusions and future work

E. coli Chemotaxis Signaling Network


Ligand binding

Methylation

Phosphorylation

CheYCheZCheZCheY

PCheBCheB

CheBTCheBT

CheYTCheYT

TT

y

b

b

y

aa

kp

kp

pEun

kEpn

Eun

kEpn

Enp

kkEun

''

40 ~

p

Fn

kBn

k

k

pFn

Fn

kRn

k

kFn

CheBTTCheBT

CheRTTCheRT

Bnc

br

bf

Rnc

rr

rf

)1(

)1(

E

nolk

lkEnv TTL

phosphorylation

methylation

Lig

an

d b

ind

ing

E=F(free form), R(coupling with CheR), B(coupling with CheBp)

E’=F(free form), R(coupling with CheR)𝜆=o(ligand occupied), v(ligand vacuum)𝛾=u(unphosphorylated), p(phosphorylated)

Michaelis-Menten Kinetics


PEESSE k

rk

fk

A key assumption in this derivation is the quasi steady state approximation, namely that the concentration of the substrate-bound enzyme changes much more slowly than those of the product and substrate. Therefore, it may be assumed that it is in steady state:

f

rm

mr

f

rf

k

kkK

K

SESE

kk

kES

ESkESkSEkdt

ESd

]][[]][[][

0][][]][[][

where Km is the Michaelis Menten Constant (MM constant)

Enzymatic reaction:

Reaction Rates


Approach …


START with a fine-tuned model of chemotaxis network that:

reproduces key features of experiments

is NOT robust

AUGMENT the model explicitly with the requirements that:

steady state value of CheY-P

values of reaction rate constants,

are independent of the external stimulus, s, thereby explicitly incorporating perfect adaptation.

s

k

F

u

skuFdt

ud

0);;(

: state variables

: reaction kinetics

: reaction rates

: external stimulus

The steady state concentration of proteins in the network satisfy:

The steady state concentration of = [CheY-P] must be independent of stimulus, s:

where parameter allows for “near-perfect” adaptation.

Reaction rates are constant and must also be independent of stimulus, s:

Augmented System


0

||

0);;(

ds

kdds

du

skuFdt

ud

N

02

|2

|

0);;(

)1(

11

11

s

kks

uu

skuFdt

ud

sjss

jm

jm

j

jN

jN

jjj

jlowj

0ds

kd

0);;( skuFdt

ud

||ds

duN

Nu

Discretize s in

range {slow, shigh}

Physical Interpretation of Parameter, : Near-Perfect Adaptation


Measurement of c = [CheY-P] by flagellar motor constrained by diffusive noise Relative accuracy*,

Signaling pathway required to adapt “nearly” perfectly, to within this lower bound

(*) Berg & Purcell, Biophys. J. (1977).

%101

~

cDac

c

: diffusion constant (~ 3 µM)

: linear dimension of motor C-ring (~ 45 nm)

: CheY-P concentration (at steady state ~ 3 µM)

: measurement time (run duration ~ 1 second)c

a

D

},,{ kuy

Use Newton-Raphson (root finding algorithm with back-tracking), to solve for the steady state of augmented system,

Use Dsode (stiff ODE solver), to verify time- dependent behavior for different ranges of external stimulus by solving:

Implementation

0

||

0);(

ds

kdds

dysyF

N

);;( skuFdt

ud


Converting from Guess to Solution


A

B

Starting from initial guess A, the solution to B is generated.

T3 autophosphorylation rate (k3a)

Inve

rse

of

T3 M

M c

on

stan

t (K

3R-1)

Parameter Surfaces


●1%<<3% ● 0%<<1%

Surface 2D projections

)(

|)()(|

beforeY

beforeYafterY

p

pp

Inve

rse

of T

1 m

eth

ylat

ion

MM

co

nsta

nt

(K

1R

-1)

Inverse of T1 demethylation MM constant(k1B

-1)

T1 autophosphorylation rate K1a

Inve

rse

of T

1 m

eth

ylat

ion

MM

co

nsta

nt

(K

1R

-1)

Validation


Time (s)

Co

nce

ntr

atio

n (

µM

)Verify steady state NR solutions dynamically using DSODE for different stimulus ramps:

Violating and Restoring Perfect Adaptation


Step stimulus from 0 to 1e-3M at t=500s

(5e+6,10)

(1e+6,10)

T3 autophosphorylation rate (k9)C

heY

p C

on

cen

trat

ion

(µ

M)

Inve

rse

of

T3 M

M c

on

stan

t (K

3R-1)

Time (s)

Conditions for Perfect Adaptation:

Kinetic Parameters

04/21/23 19Yang Yang, Candidacy Seminar

Inverse of Methylation MM Constant Autophosphorylation Rate



Inve

rse

of

T0 M

M

con

stan

t (K

0R-1)


Inve

rse

of

T1 M

M

con

stan

t (K

1 R-1)





Inve

rse

of

T2 M

M

con

stan

t (K

2R-1)

Inve

rse

of

T3 M

M

con

stan

t (K

3R-1)



LT0 autophosphorylation rate (k0al)


Inve

rse

of

LT

0 M

M

con

stan

t (K

0LR

-1)

Inve

rse

of

LT

1 M

M

con

stan

t (K

1LR

-1)





Inve

rse

of

LT

2 M

M

con

stan

t (K

2LR

-1)

Inve

rse

of

LT

3 M

M

con

stan

t (K

3LR

-1)

Inverse of Demethylation MM Constant Autophosphorylation Rate




Inve

rse

of

T1 M

M

con

stan

t (K

1B-1)

Inve

rse

of

T2

MM

co

nst

ant

(K2B

-1)





Inve

rse

of

T3 M

M

con

stan

t (K

3B-1)

Inve

rse

of

T4

MIM

co

nst

ant

(K4B

-1)



LT1 autophosphorylation rate (k1al) LT2 autophosphorylation rate (k2al)

Inve

rse

of

LT

1 M

M

con

stan

t (K

1LB

-1)

Inve

rse

of

LT

2 M

M

con

stan

t (K

2LB

-1)



LT3 autophosphorylation rate (k12) LT4 autophosphorylation rate (k13)

Inve

rse

of

LT

3 M

M

con

stan

t (K

2LB

-1)

Inve

rse

of

LT

4 M

M

con

stan

t (K

3LB

-1)

Methylation Catalytic Rate/Demethylation Catalytic Rate = Constant


T1 demethylation catalytic rate

T1

met

hyl

atio

n c

atal

ytic

rat

e


T2

met

hyl

atio

n c

atal

ytic

rat

e




T2

met

hyl

atio

n c

atal

ytic

rat

e


T3

met

hyl

atio

n c

atal

ytic

rat

e



LT1 demethylation catalytic rate

LT

0 m

eth

ylat

ion

cat

alyt

ic

rate


LT

1 m

eth

ylat

ion

cat

alyt

ic

rate

Methylation Catalytic Rate/Demethylation Catlytic Rate = Constant



LT

2 d

emet

hyl

atio

n c

atal

ytic

ra

te


LT

3 d

emet

hyl

atio

n c

atal

ytic

ra

te

Summary


These conditions are consistent with those obtained in previous works from analysis of a detailed, two-state receptor model*.

The Inverse of Methylation MM constants linearly

decrease with Autophosphorylation RatesThe Inverse of Demethylation MM constants linearly

increase with Autophosphorylation RatesThe ratio of Methylation catalytic rates and demethylation

catlytic rates for the next methylation level is constant for all

methylation states

* B. Mello et al. Biophysical Journal , (2003).

Some Conditions in Two-State Receptor Model


These conditions are consistent with those obtained in previous works from analysis of a detailed, two-state receptor model*.

The Inverse of Methylation MM constants linearly

decrease with Autophosphorylation RatesThe Inverse of Demethylation MM constants linearly

increase with Autophosphorylation RatesThe ratio of Methylation catalytic rates and demethylation

catlytic rates for the next methylation level is constant for all

methylation states

* B. Mello et al. Biophysical Journal , (2003).

Conditions for Perfect Adaptation:

Protein Concentrations

Summary of Protein Concentrations


Relationship Between Protein Concentrations


(M)

(M)

(M)(M)

Relationship Between Protein Concentrations (cont’d)


(M)

(M)

(M)

(M)

Relationship between Protein Concentrations (cont’d)


(M)

(M)

(M)

(M)

Diversity of Chemotaxis Systems


Eg., Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteria possess multiple CheYs while lacking of CheZ homologue.

In different bacteria, additional protein components as well as multiple copies of certain chemotaxis proteins are present.

Response regulator

Phosphate “sink”

CheY1CheY2

Two CheY System


Exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ:

Ch

eY1

p (µ

M)

Ch

eY1

p (µ

M)

Time(s) Time(s)

Requiring: Faster phosphorylation/autodephosphorylation rates of CheY2 than CheY1

Faster phosphorylation rate of CheB

Conclusions


I. Successful implementation of a novel method for elucidating regions in parameter space allowing precise adaptation

II. Numerical results for (near-) perfect adaptation manifolds in parameter space for the E. coli chemotaxis network, allowing determination of

i. Conditions required for perfect adaptation, consistent with and extending previous works [1-3]

ii. Numerical ranges for experimentally unknown or partially known kinetic parameters

I. Extension to modified chemotaxis networks, for example with no CheZ homologue and multiple CheYs

[1] Barkai & Leibler, Nature (1997). [2] Yi et al., PNAS (2000). [3] Tu & Mello, Biophys. J. (2003).

Future Work


Extension to other signaling networks

vertebrate phototransduction mammalian circadian clock

allowing determination of

a) parameter dependences underlying robustness of adaptation

b) plausible numerical values for unknown network parameters

Vertebrate Phototransduction


http://www.fz-juelich.de/inb/inb-1/Photoreception/

•cGMP: cyclic GMP

•PDE: cGMP phosphodiesterase

•GCAP: guanylyl cyclase

activating, Ca2+ binding protein

•gc: guanylyl cyclase, which

synthesis cGMP

GCAPgccGMPGMPGCAPgc

GCAPgcgcGCAP

CaGCAPCaGCAP

PDEGMPcGMPPDE

RhPDEPDERh

pRhRhp

**

22

**

***

*

*

Light Adaptation of Phototransduction


An intracellular recording from a single cone stimulated with different amounts of light. Each trace represents the response to a brief flash that was varied in intensity. At the highest light levels, the response amplitude saturates. (Neuroscience, Purves et al., 2001)

Kinetic Model for Vertebrate Phototransduction


Russell D. Hamer, Visual Neuroscience (2000)

Mammalian Circadian Clock


http://www.umassmed.edu/neuroscience/faculty/reppert.cfm?start=0

PERs transport CRYs to nucleusCLOCK and BMAL1 bind togetherCLOCK·BMAL1 binds to E box to increase Pers(Crys) transcription ratesE box is the sequence CACGTG of the PER1 and CRY1 genes PERs bind with kinases CKIε/δ to be phosphorylatedPhosphorylated PERs bind with CRYsOnly phosphorylated PER·CRY· CKIε/δ can enter nucleusPhosphorylated PER·CRY· CKIε/δ inhibit the ability of CLOCK·BMALI to enhance transcriptionIncreasing REV-ERBα levels repress BMAL1 transcriptionActivator positively regulated BMAL1 transcription

From Forger et al., PNAS (2003).


A

B

C

D




inve

rse

of

T2 M

M

con

stan

t (K

2R-1)

inve

rse

of

T3 M

M

con

stan

t (K

3R-1)




inve

rse

of

T1 M

-M

con

stan

t (K

1B-1)

inve

rse

of

T2

M-M

co

nst

ant

(K2B

-1)




inve

rse

of

T3 M

-M

con

stan

t (K

3B-1)

inve

rse

of

T4

M-M

co

nst

ant

(K4B

-1)


LT1 autophosphorylation rate (k1al) LT2 autophosphorylation rate (k2al)

inve

rse

of

LT

1 M

M

con

stan

t (K

1LB

-1)

inve

rse

of

LT

2 M

M

con

stan

t (K

2LB

-1)


LT3 autophosphorylation rate (k12) LT4 autophosphorylation rate (k13)

inve

rse

of

LT

3 M

M

con

stan

t (K

2LB

-1)

inve

rse

of

LT

4 M

M

con

stan

t (K

3LB

-1)

near-perfect adaptation in bacterial chemotaxis

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