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description
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
04/21/23 Yang Yang, Candidacy Seminor 3
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
04/21/23 Yang Yang, Candidacy Seminor 4
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
04/21/23 Yang Yang, Candidacy Seminor 5
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
04/21/23 Yang Yang, Candidacy Seminor 7
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
04/21/23 8Yang Yang, Candidacy Seminor
Augmented SystemAugmented System
04/21/23 Yang Yang, Candidacy Seminor 9
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
04/21/23 Yang Yang, Candidacy Seminor 10
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
04/21/23 11Yang Yang, Candidacy Seminor
Michaelis Menten kinetics and Michaelis Menten kinetics and constantsconstants
04/21/23 Yang Yang, Candidacy Seminor 12
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
04/21/23 Yang Yang, Candidacy Seminor 13
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:
04/21/23 15Yang Yang, Candidacy Seminor
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
04/21/23 Yang Yang, Candidacy Seminor 17
Conditions for (Near-)Perfect Adaptation
Inverse of Methylation MM constant Inverse of Methylation MM constant Autophosphorylation RateAutophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminor 18
T0 autophosphorylation rate (k0a)
inve
rse
of
T0 M
-M
con
stan
t (K
0R-1)
T1 autophosphorylation rate (k1a)
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)
04/21/23 Yang Yang, Candidacy Seminor 19
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)
Inverse of Methylation MM constant Inverse of Methylation MM constant Autophosphorylation Rate(cont’d)Autophosphorylation Rate(cont’d)
04/21/23 Yang Yang, Candidacy Seminor 20
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)
Inverse of Methylation MM constant Inverse of Methylation MM constant Autophosphorylation Rate(cont’d)Autophosphorylation Rate(cont’d)
04/21/23 Yang Yang, Candidacy Seminor 21
LT2 autophosphorylation rate (k2al) LT3 autophosphorylation rate (k3al)
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
04/21/23 Yang Yang, Candidacy Seminor 22
T1 autophosphorylation rate (k1a) T2 autophosphorylation rate (k2a)
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)
04/21/23 Yang Yang, Candidacy Seminor 23
T3 autophosphorylation rate (k3a) T4 autophosphorylation rate (k4a)
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)
Inverse of Demethylation MM constant Inverse of Demethylation MM constant Autophosphorylation Rate(cont’d)Autophosphorylation Rate(cont’d)
04/21/23 Yang Yang, Candidacy Seminor 24
LT1 autophosphorylation rate (k1al) LT2 autophosphorylation rate (k2al)
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)
Inverse of Demethylation MM constant Inverse of Demethylation MM constant Autophosphorylation Rate(cont’d)Autophosphorylation Rate(cont’d)
04/21/23 Yang Yang, Candidacy Seminor 25
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
04/21/23 Yang Yang, Candidacy Seminor 26
T1 demethylation catalytic rate
T 1 met
hyla
tion
cata
lytic
rate
T2 demethylation catalytic rate
T 2 met
hyla
tion
cata
lytic
rate
Methylation catalytic rate/Methylation catalytic rate/demethylation catlytic rate is constantdemethylation catlytic rate is constant
04/21/23 Yang Yang, Candidacy Seminor 27
T3 demethylation catalytic rate
T 2 met
hyla
tion
cata
lytic
rate
T4 demethylation catalytic rate
T 3 met
hyla
tion
cata
lytic
rate
04/21/23 Yang Yang, Candidacy Seminor 28
LT1 demethylation catalytic rate
LT0 m
ethy
latio
n ca
taly
tic ra
te
LT2 demethylation catalytic rate
LT1 m
ethy
latio
n ca
taly
tic ra
te
Methylation catalytic rate/Methylation catalytic rate/demethylation catlytic rate is constantdemethylation catlytic rate is constant
04/21/23 Yang Yang, Candidacy Seminor 29
LT3 demethylation catalytic rate
LT2 d
emet
hyla
tion
cata
lytic
rate
LT4 demethylation catalytic rate
LT3 d
emet
hyla
tion
cata
lytic
rate
Methylation catalytic rate/Methylation catalytic rate/demethylation catlytic rate is constantdemethylation catlytic rate is constant
SummarySummary
04/21/23 Yang Yang, Candidacy Seminor 30
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
04/21/23 Yang Yang, Candidacy Seminor 31
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
04/21/23 Yang Yang, Candidacy Seminor 32
Conditions of protein concentrations for (Near-)
Perfect Adaptation
Protein concentrationsProtein concentrations
04/21/23 Yang Yang, Candidacy Seminor 33
Relationship between protein Relationship between protein concentrations concentrations
04/21/23 Yang Yang, Candidacy Seminor 34
(M)
(M)
(M)
(M)
Relationship between protein Relationship between protein concentrations concentrations
04/21/23 Yang Yang, Candidacy Seminor 35
(M)
(M)
(M)
(M)
Relationship between protein Relationship between protein concentrations concentrations
04/21/23 Yang Yang, Candidacy Seminor 36
(M)
(M)
(M)
(M)
Diversity of Chemotaxis SystemsDiversity of Chemotaxis Systems
04/21/23 Yang Yang, Candidacy Seminor 37
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
04/21/23 Yang Yang, Candidacy Seminor 38
Exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ:
Conclusions Conclusions
04/21/23 Yang Yang, Candidacy Seminor 39
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
04/21/23 Yang Yang, Candidacy Seminor 40
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
04/21/23 Yang Yang, Candidacy Seminor 41
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
04/21/23 Yang Yang, Candidacy Seminor 42
J. M. Valkkin and Dirk Van Norren, Vision Res. (1983).
Differential equations for verterbrate Differential equations for verterbrate phototransduction phototransduction
04/21/23 Yang Yang, Candidacy Seminor 43
Russell D. Hamer, Visual Neuroscience (2000)
Mammalian circadian clockMammalian circadian clock
04/21/23 Yang Yang, Candidacy Seminor 44
http://www.umassmed.edu/neuroscience/faculty/reppert.cfm?start=0
04/21/23 Yang Yang, Candidacy Seminor 45
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