12/24/2015yang yang, candidacy seminar1 near-perfect adaptation in bacterial chemotaxis yang yang...

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 as a Model Organism


Workhorse of molecular biology,most studied cell in all science:

- Small genome (~4300 genes) - Normal lack of pathogenicity - Ease of growth in lab

Basis of recent developments in biotechnology and genetic engineering, including living factory for producing human medicines

Basis for understanding of fundamental cellular processes: - cellular sensory systems - regulation of gene expression - cell division, etc.

Dimensions: Body size: 1 μm in length

0.4 μm in radius Flagellum: 10 μm long

Cell cycle:~ 1 hour

E. coli in Motion


QuickTime™ and aMPEG-4 Video decompressor

are needed to see this picture.

From Berg & Brown, Nature (1972).

Physical constants: Cell speed: 20 - 30 μm/sec

Mean run time: 1 secMean tumble time: 0.1 sec

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

Increasing attractants or Decreasing repellents

Bacterial Chemotaxis

• “Hydrogen atom” of biochemical signal transduction networks

• Paradigm for two-component receptor-regulated phosphorylation pathways

• Accessible for study by structural, biochemical and genetic approaches

The chemosensory pathway in bacterial chemotaxis and propulsion system it regulates have provided an ideal system for probing the physical principles governing complex cellular signaling and response.

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

Flagellar Motor in E. coli


From R. M. Berry, Encyclopedia of Life Science (2001).

From P. Cluzel, et al., Science (2000).

Response to Step Stimulus


Fast response Slow adaptation

From Block et al., Cell (1982).

From Sourjik et al., PNAS (2002).

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

CheY-P response:Flagellar response:

Excitation and Adaptation


Precision of Adaptation


From Alon et al. Nature (1999).

Precision of adaptation =steady state tumbling frequency of unstimulated cells /steady state tumbling frequency of stimulated cells

Squares: Unstimulated cellsCircles: Cells stimulated at t=0

(Each point represents data from 10s motion of 100-400 cells.)

Robustness of Perfect Adaptation


From Alon et al. Nature (1999).

Precision of adaptation robust to 50-fold change in CheR expression …

…while …

Adaptation time and steady state tumbling frequency vary significantly.

Robustness of perfect adaptation: precision of adaptation insenstive to network parameters

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)

Enzymatic reaction:

Where E is the enzyme, E0 is the total enzyme concentration, S is the substrate, P is the product.

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:

Michaelis-Menten Kinetics


PEESSE k

rk

fk

f

rm

mm

mr

f

rf

k

kkK

SK

SV

SK

SEk

dt

dP

EESE

K

SESE

kk

kES

ESkESkSEkdt

ESd

][

][

][

][][

][][][

]][[]][[][

0][][]][[][

max0

0

where Km is the Michaelis-Menten (MM) constant

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


Convergence from Guess to Solution


A

B

Starting from initial guess A, solution B is generated. By comprehensively sampling space of parameters with initial guesses, solution “surfaces” are constructed.

T3 autophosphorylation rate (k3a)

Inve

rse

of

T3 M

M c

on

stan

t (K

3R-1)

●3%<<5% ●1%<<3% ● 0%<<1%

)(

|)()(|

beforeY

beforeYafterY

p

pp

ds

Y

ds

Y

Y

dY

ds

dY pp

p

pp

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)

Slices of 3D Surfaces of Parameter Space


1 2 3 4

5 6 7 8

9 10 11

1 -11 denote slices perpendicular to K1B

-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 (k3a)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 24Yang 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

1R-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 (k3al) LT4 autophosphorylation rate (k4al)

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 k1b

T0

met

hyl

atio

n c

atal

ytic

rat

e k 0

c


T1

met

hyl

atio

n c

atal

ytic

rat

e k 1

c




T2

met

hyl

atio

n c

atal

ytic

rat

e k 2

c


T3

met

hyl

atio

n c

atal

ytic

rat

e k 3

c



LT1 demethylation catalytic rate k1bl

LT

0 m

eth

ylat

ion

cat

alyt

ic

rate

k0c

l


LT

1 m

eth

ylat

ion

cat

alyt

ic

rate

k1c

l

Methylation Catalytic Rate/Demethylation Catlytic Rate = Constant



LT

2 d

emet

hyl

atio

n c

atal

ytic

ra

te k

2cl


LT

3 d

emet

hyl

atio

n c

atal

ytic

ra

te k

3cl

Summary: Reaction Kinetics


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

Inverse of methylation MM constants linearly decreases

with autophosphorylation ratesInverse of demethylation MM constants linearly increases

with autophosphorylation ratesRatio of methylation catalytic rates and demethylation

catalytic 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

Intrinsic Variability in Total Protein Concentrations


Total chemotaxis protein concentrations vary across clonal population of cells, as well as across cell cycles due to fluctuations in gene expression and partitioning of proteins at cell division.

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)

Summary: Protein Concentrations


Preliminary observations :• Total receptor concentration and CheY concentration

have a lower and upper boundary respectly.• CheR concentration is proportional to the CheB

concentration

CheR, CheB restricted ranges consistent with experiment*.

* Li and Hazelbauer, Journal of Bacteriology , (2004).

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

Example: 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

iii. Preliminary results on restrictions of total protein concentrations

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 Extensions


Extension to other signaling networks

- vertebrate phototransduction [1]

- mammalian circadian clock [2]

allowing determination of

a) parameter dependences underlying robustness of adaptation

b) plausible numerical values for unknown network parameters

[1] R. D. Hamer, et al., Vis Neurosci 22, 417-36 (2005).[2] D. B. Forger and C. S. Peskin, PNAS 102, 321-24 (2005).

Vertebrate Phototransduction


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

•cGMP: cyclic GMP

•PDE: cGMP phosphodiesterase

•GCAP: guanylyl cyclase

•gc: guanylyl cyclase

GCAPgccGMPGMPGCAPgc

GCAPgcgcGCAP

CaGCAPCaGCAP

PDEGMPcGMPPDE

RhPDEPDERh

pRhRhp

**

22

**

***

*

*

Light Adaptation in 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 (From 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).

Conditions 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 of the activity (hence, 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 phospho-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(

Checking Dynamics of CheY-P with Solutions


A

B

C

D

Protein Concentration Trend Shifting


Reaction Rates Trend Shifting




inve

rse

of

T2 M

M c

on

stan

t (K

2R-1)

inve

rse

of

T3 M

M c

on

stan

t (K

3R-1)

Protein concentrations taken from SPO’s

Protein concentrations taken from Mello-Tu’s





inve

rse

of

T2 M

M c

on

stan

t (K

2R-1)

inve

rse

of

T3 M

M c

on

stan

t (K

3R-1)Protein concentrations

taken from SPO’s






inve

rse

of

T1 M

-M c

on

stan

t (K

1B-1)

inve

rse

of

T2

M-M

co

nst

ant

(K2B

-1)Protein concentrations

taken from SPO’s






inve

rse

of

T3 M

-M c

on

stan

t (K

3B-1)

inve

rse

of

T4

M-M

co

nst

ant

(K4B

-1)







inve

rse

of

LT

1 M

M c

on

stan

t (K

1LB

-1)

inve

rse

of

LT

2 M

M c

on

stan

t (K

2LB

-1)





LT3 autophosphorylation rate (k12)

LT4 autophosphorylation rate (k13)

inve

rse

of

LT

3 M

M c

on

stan

t (K

2LB

-1)

inve

rse

of

LT

4 M

M c

on

stan

t (K

3LB

-1)





1 2 3 4

5 6 7 8

9 10 111------------------------11



1 2 3 4

5 6 7 8

9 10 11

1----------------------11

Slices of 3D Surfaces of Parameter Space Comparing Pair-Wise Relationship



Inv

ers

e o

f T

1 M

M c

on

sta

nt

(K1

R-1)


Inv

ers

e o

f T

1 M

M c

on

sta

nt

(K1B

-1)

E. coli and Bacteria Chemotaxis


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

Increasing attractants or Decreasing repellents

12/24/2015yang yang, candidacy seminar1 near-perfect adaptation in bacterial chemotaxis yang yang...

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