Near-Perfect Adaptation in Bacterial Chemotaxis

Near-Perfect Adaptation Near-Perfect Adaptation in Bacterial Chemotaxisin Bacterial Chemotaxis

Yang Yang

Advisor: Sima Setayeshgar

Department of Physics

Indiana University, Bloomington, IN

04/21/23 1Yang Yang, Candidacy Seminor

E. coli E. coli and Bacteria Chemotaxisand Bacteria Chemotaxis

04/21/23 Yang Yang, Candidacy Seminor 2

Increasing attractants or Decreasing repellents

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

Chemotaxis Signal Transduction Chemotaxis Signal Transduction Network in Network in E. coliE. coli


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 AdaptationRobust Perfect Adaptation


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)

This Work: OutlineThis 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

Modified fine-tuned modelModified fine-tuned model

04/21/23 Yang Yang, Candidacy Seminor6

Ligand binding

Methylation

Phosphorylation

CheYCheZCheZCheY

PCheBCheB

CheBTCheBT

CheYTCheYT

TT

y

b

b

y

aa

kp

kp

pE

unkE

pn

Eun

kEpn

Enp

kkEun

''

40 ~

p

Fn

kBn

k

k

pF

n

Fn

kRn

k

kF

n

CheBTTCheBT

CheRTTCheRT

Bnc

br

bf

Rnc

rr

rf

)1(

)1(

E

nolk

lkE

nv TTL

phosphorylation

methylation

Liga

nd b

indi

ng

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)

Reaction ratesReaction rates


Approach …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


Augmented SystemAugmented System


The steady state concentration of proteins in the network satisfy:

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

where parameter ε allows for “near-perfect” adaptation.

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

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 Discretize s in

range {slow, shigh}

Physical Interpretation of Physical Interpretation of εε : : Near-Perfect adaptationNear-Perfect adaptation


Measurement of c = [CheY-P] by flagella 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:

ImplementationImplementation

0

||

0);(

ds

kdds

dysyF

N

);;( skuFdt

ud


Michaelis Menten kinetics and Michaelis Menten kinetics and constantsconstants


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 change much more slowly than those of the product and substrate and we can assume it is always in steady state, then:

f

rm

mr

f

rf

k

kkK

K

SESE

kk

kES

ESkESkSEkdt

ESd

]][[]][[][

0][][]][[][

Where Km is the Michaelis Menten Constant(MM constant)

A chemical reaction:

Converting from guess to solutionConverting from guess to solution


A

B

Starting from initial guess A, the solver converted the solution to B

T3 autophosphorylation rate (k3a)

inve

rse

of

T3 M

-M c

on

stan

t (K

3R-1)

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

Parameter SurfacesParameter Surfaces

●1%<<3% ● 0%<<1%

Surface 2D projections

)(

|)()(|

beforeY

beforeYafterY

p

pp

04/21/23 14Yang Yang, Candidacy Seminork

Inve

rse

of T

1 m

eth

ylat

ion

MM

co

nsta

nt(

k 1R

-1)

ValidationValidation

Time (s)

Conc

entr

ation

(µM

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


Violating and Restoring Violating and Restoring Perfect AdaptationPerfect Adaptation

04/21/23 Yang Yang, Candidacy Seminor 16 Step stimulus from 0 to 1e-3M at t=500s

(5e+6,10)

(1e+6,10)

Time (s)

CheY

p Co

ncen

trati

on (µ

M)

15%

2%

T3 autophosphorylation rate (k9)

inve

rse

of

T3 M

-M c

on

stan

t (K

3R-1)

ResultsResults


Conditions for (Near-)Perfect Adaptation

Inverse of Methylation MM constant Inverse of Methylation MM constant Autophosphorylation RateAutophosphorylation 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)

Inverse of Methylation MM constant Inverse of Methylation MM constant Autophosphorylation Rate(cont’d)Autophosphorylation Rate(cont’d)


T2 autophosphorylation rate (k2a) T3 autophosphorylation rate (k3a)

in

vers

e o

f T

2 M

M

con

stan

t (K

2R-1)

inve

rse

of

T3 M

M

con

stan

t (K

3R-1)



LT0 autophosphorylation rate (k0al) LT1 autophosphorylation rate (k1al)

in

vers

e o

f L

T0 M

M

con

stan

t (K

0LR

-1)

inve

rse

of

LT

1 M

M

con

stan

t (K

1LR

-1)




in

vers

e o

f L

T2 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 Inverse of Demethylation MM constant Autophosphorylation RateAutophosphorylation Rate



in

vers

e o

f T

1 M

-M

con

stan

t (K

1B-1)

inve

rse

of

T2

M-M

co

nst

ant

(K2B

-1)

Inverse of Demethylation MM constant Inverse of Demethylation MM constant Autophosphorylation Rate(cont’d)Autophosphorylation Rate(cont’d)



in

vers

e o

f T

3 M

-M

con

stan

t (K

3B-1)

inve

rse

of

T4

M-M

co

nst

ant

(K4B

-1)




in

vers

e o

f L

T1 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)

in

vers

e o

f L

T3 M

M

con

stan

t (K

2LB

-1)

inve

rse

of

LT

4 M

M

con

stan

t (K

3LB

-1)

Methylation catalytic rate/Methylation catalytic rate/demethylation catlytic rate is constantdemethylation catlytic rate is constant


T1 demethylation catalytic rate

T 1 met

hyla

tion

cata

lytic

rate


T 2 met

hyla

tion

cata

lytic

rate




T 2 met

hyla

tion

cata

lytic

rate


T 3 met

hyla

tion

cata

lytic

rate


LT1 demethylation catalytic rate

LT0 m

ethy

latio

n ca

taly

tic ra

te


LT1 m

ethy

latio

n ca

taly

tic ra

te




LT2 d

emet

hyla

tion

cata

lytic

rate


LT3 d

emet

hyla

tion

cata

lytic

rate


SummarySummary


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 in two-state receptor modelConditions in two-state receptor model


Receptor autophosphorylation rates are proportional to the receptor activity:

Only the inactive or active receptors can be methylated or demethylated. The association rates between receptors and CheR or CheBp are linearly

related to the receptor activity, while dissociation rates are independent with 𝜆. Then the inverse of the methylation or demethylation MM constants are linearly related to the receptor activity:

The ratios between methylation catalytic rates and demethylation catalytic rates for the next methylation level are constant:

The phosphate transfer rates from CheA to CheB or CheY are proportional to receptor activities:

nan Pk

nBnn

Rn PKPK 11 )(,1)(

nPBnn

PYn PkPk ,

constantk

kRn

Bn

)1(

ResultsResults


Conditions of protein concentrations for (Near-)

Perfect Adaptation

Protein concentrationsProtein concentrations


Relationship between protein Relationship between protein concentrations concentrations


(M)

(M)

(M)

(M)



(M)

(M)

(M)

(M)

Diversity of Chemotaxis SystemsDiversity 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

Requiring: Faster phosphorylation/autodephosphorylation rates of CheY2 than CheY1

Faster phosphorylation rate of CheB

CheY

1p (µ

M)

CheY

1p (µ

M)

Time(s)

Two CheY SystemTwo CheY System


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

Conclusions 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

ii. numerical ranges for unknown or partially known kinetic parameters

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

Future WorkFuture Work


Extension to other signaling networks

vertebrate phototransduction mammalian circadian clock

allowing determination of

a) parameter dependences underlying robustness

b)plausible numerical values for unknown network parameters

vertebrate phototransductionvertebrate 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

cGMPGMPgc

gcgcGCAP

CaGCAPCaGCAP

GMPcGMPPDE

PDEPDERh

RhRhp

*

22

*

**

*

*

Light adaptation of primate conesLight adaptation of primate cones


J. M. Valkkin and Dirk Van Norren, Vision Res. (1983).

Differential equations for verterbrate Differential equations for verterbrate phototransduction phototransduction


Russell D. Hamer, Visual Neuroscience (2000)

Mammalian circadian clockMammalian circadian clock


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

Near-Perfect Adaptation in Bacterial Chemotaxis

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