near-perfect adaptation in bacterial chemotaxis
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
04/21/23 Yang Yang, Candidacy Seminar 4
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
04/21/23 Yang Yang, Candidacy Seminar 7
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
04/21/23 Yang Yang, Candidacy Seminar 8
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
04/21/23 Yang Yang, Candidacy Seminar 9
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
04/21/23 Yang Yang, Candidacy Seminar 10
Approach …
04/21/23 Yang Yang, Candidacy Seminar 11
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
04/21/23 Yang Yang, Candidacy Seminar 12
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
04/21/23 Yang Yang, Candidacy Seminar 13
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
04/21/23 Yang Yang, Candidacy Seminar 14
Converting from Guess to Solution
04/21/23 Yang Yang, Candidacy Seminar 15
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
04/21/23 Yang Yang, Candidacy Seminar 16
●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
04/21/23 Yang Yang, Candidacy Seminar 17
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
04/21/23 Yang Yang, Candidacy Seminar 18
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
04/21/23 Yang Yang, Candidacy Seminar 20
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 Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 21
T2 autophosphorylation rate (k2a)
T3 autophosphorylation rate (k3a)
Inve
rse
of
T2 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 Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 22
LT0 autophosphorylation rate (k0al)
LT1 autophosphorylation rate (k1al)
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)
Inverse of Methylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 23
LT2 autophosphorylation rate (k2al)
LT3 autophosphorylation rate (k3al)
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
04/21/23 Yang Yang, Candidacy Seminar 24
T1 autophosphorylation rate (k1a)
T2 autophosphorylation rate (k2a)
Inve
rse
of
T1 M
M
con
stan
t (K
1B-1)
Inve
rse
of
T2
MM
co
nst
ant
(K2B
-1)
Inverse of Demethylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 25
T3 autophosphorylation rate (k3a)
T4 autophosphorylation rate (k4a)
Inve
rse
of
T3 M
M
con
stan
t (K
3B-1)
Inve
rse
of
T4
MIM
co
nst
ant
(K4B
-1)
Inverse of Demethylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 26
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)
Inverse of Demethylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 27
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
04/21/23 Yang Yang, Candidacy Seminar 28
T1 demethylation catalytic rate
T1
met
hyl
atio
n c
atal
ytic
rat
e
T2 demethylation catalytic rate
T2
met
hyl
atio
n c
atal
ytic
rat
e
Methylation Catalytic Rate/Demethylation Catalytic Rate = Constant
04/21/23 Yang Yang, Candidacy Seminar 29
T3 demethylation catalytic rate
T2
met
hyl
atio
n c
atal
ytic
rat
e
T4 demethylation catalytic rate
T3
met
hyl
atio
n c
atal
ytic
rat
e
Methylation Catalytic Rate/Demethylation Catalytic Rate = Constant
04/21/23 Yang Yang, Candidacy Seminar 30
LT1 demethylation catalytic rate
LT
0 m
eth
ylat
ion
cat
alyt
ic
rate
LT2 demethylation catalytic rate
LT
1 m
eth
ylat
ion
cat
alyt
ic
rate
Methylation Catalytic Rate/Demethylation Catlytic Rate = Constant
04/21/23 Yang Yang, Candidacy Seminar 31
LT3 demethylation catalytic rate
LT
2 d
emet
hyl
atio
n c
atal
ytic
ra
te
LT4 demethylation catalytic rate
LT
3 d
emet
hyl
atio
n c
atal
ytic
ra
te
Summary
04/21/23 Yang Yang, Candidacy Seminar 32
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
04/21/23 Yang Yang, Candidacy Seminar 33
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
04/21/23 Yang Yang, Candidacy Seminar 35
Relationship Between Protein Concentrations
04/21/23 Yang Yang, Candidacy Seminar 36
(M)
(M)
(M)(M)
Relationship Between Protein Concentrations (cont’d)
04/21/23 Yang Yang, Candidacy Seminar 37
(M)
(M)
(M)
(M)
Relationship between Protein Concentrations (cont’d)
04/21/23 Yang Yang, Candidacy Seminar 38
(M)
(M)
(M)
(M)
Diversity of Chemotaxis Systems
04/21/23 Yang Yang, Candidacy Seminar 40
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
04/21/23 Yang Yang, Candidacy Seminar 41
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
04/21/23 Yang Yang, Candidacy Seminar 42
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
04/21/23 Yang Yang, Candidacy Seminar 43
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
04/21/23 Yang Yang, Candidacy Seminar 44
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
04/21/23 Yang Yang, Candidacy Seminar 45
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
04/21/23 Yang Yang, Candidacy Seminar 46
Russell D. Hamer, Visual Neuroscience (2000)
Mammalian Circadian Clock
04/21/23 Yang Yang, Candidacy Seminar 47
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).
04/21/23 Yang Yang, Candidacy Seminar 48
04/21/23 Yang Yang, Candidacy Seminar 49
04/21/23 Yang Yang, Candidacy Seminar 50
04/21/23 Yang Yang, Candidacy Seminar 51
04/21/23 Yang Yang, Candidacy Seminar 52
04/21/23 Yang Yang, Candidacy Seminar 53
04/21/23 Yang Yang, Candidacy Seminar 54
A
B
C
D
04/21/23 Yang Yang, Candidacy Seminar 55
04/21/23 Yang Yang, Candidacy Seminar 56
04/21/23 Yang Yang, Candidacy Seminar 57
04/21/23 Yang Yang, Candidacy Seminar 58
04/21/23 Yang Yang, Candidacy Seminar 59
04/21/23 Yang Yang, Candidacy Seminar 60
T2 autophosphorylation rate (k2a)
T3 autophosphorylation rate (k3a)
inve
rse
of
T2 M
M
con
stan
t (K
2R-1)
inve
rse
of
T3 M
M
con
stan
t (K
3R-1)
04/21/23 Yang Yang, Candidacy Seminar 61
T2 autophosphorylation rate (k2a)
T3 autophosphorylation rate (k3a)
inve
rse
of
T2 M
M
con
stan
t (K
2R-1)
inve
rse
of
T3 M
M
con
stan
t (K
3R-1)
04/21/23 Yang Yang, Candidacy Seminar 62
T1 autophosphorylation rate (k1a)
T2 autophosphorylation rate (k2a)
inve
rse
of
T1 M
-M
con
stan
t (K
1B-1)
inve
rse
of
T2
M-M
co
nst
ant
(K2B
-1)
04/21/23 Yang Yang, Candidacy Seminar 63
T3 autophosphorylation rate (k3a)
T4 autophosphorylation rate (k4a)
inve
rse
of
T3 M
-M
con
stan
t (K
3B-1)
inve
rse
of
T4
M-M
co
nst
ant
(K4B
-1)
04/21/23 Yang Yang, Candidacy Seminar 64
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)
04/21/23 Yang Yang, Candidacy Seminar 65
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)