12/24/2015yang yang, candidacy seminar1 near-perfect adaptation in bacterial chemotaxis yang yang...
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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
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E. Coli as a Model Organism
04/21/23 Yang Yang, Candidacy Seminar 2
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
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E. coli in Motion
04/21/23 Yang Yang, Candidacy Seminar 3
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
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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.
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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
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Flagellar Motor in E. coli
04/21/23 Yang Yang, Candidacy Seminar 6
From R. M. Berry, Encyclopedia of Life Science (2001).
From P. Cluzel, et al., Science (2000).
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Response to Step Stimulus
04/21/23 Yang Yang, Candidacy Seminar 7
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:
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Excitation and Adaptation
04/21/23 Yang Yang, Candidacy Seminar 8
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Precision of Adaptation
04/21/23 Yang Yang, Candidacy Seminar 9
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.)
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Robustness of Perfect Adaptation
04/21/23 Yang Yang, Candidacy Seminar 10
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
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This Work: Outline
04/21/23 Yang Yang, Candidacy Seminar 11
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
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E. coli Chemotaxis Signaling Network
04/21/23 Yang Yang, Candidacy Seminar 12
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)
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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
04/21/23 Yang Yang, Candidacy Seminar 13
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
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Reaction Rates
04/21/23 Yang Yang, Candidacy Seminar 14
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Approach …
04/21/23 Yang Yang, Candidacy Seminar 15
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
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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 16
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}
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Physical Interpretation of Parameter, : Near-Perfect Adaptation
04/21/23 Yang Yang, Candidacy Seminar 17
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
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},,{ 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 18
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Convergence from Guess to Solution
04/21/23 Yang Yang, Candidacy Seminar 19
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
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Parameter Surfaces
04/21/23 Yang Yang, Candidacy Seminar 20
●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)
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Slices of 3D Surfaces of Parameter Space
04/21/23 Yang Yang, Candidacy Seminar 21
1 2 3 4
5 6 7 8
9 10 11
1 -11 denote slices perpendicular to K1B
-1
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Validation
04/21/23 Yang Yang, Candidacy Seminar 22
Time (s)
Co
nce
ntr
atio
n (
µM
)Verify steady state NR solutions dynamically using DSODE for different stimulus ramps:
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Violating and Restoring Perfect Adaptation
04/21/23 Yang Yang, Candidacy Seminar 23
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)
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Conditions for Perfect Adaptation:
Kinetic Parameters
04/21/23 24Yang Yang, Candidacy Seminar
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Inverse of Methylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 25
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
1R-1)
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Inverse of Methylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 26
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)
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Inverse of Methylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 27
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)
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Inverse of Methylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 28
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)
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Inverse of Demethylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 29
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)
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Inverse of Demethylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 30
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)
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Inverse of Demethylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 31
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)
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Inverse of Demethylation MM Constant Autophosphorylation Rate
04/21/23 Yang Yang, Candidacy Seminar 32
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)
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Methylation Catalytic Rate/Demethylation Catalytic Rate = Constant
04/21/23 Yang Yang, Candidacy Seminar 33
T1 demethylation catalytic rate k1b
T0
met
hyl
atio
n c
atal
ytic
rat
e k 0
c
T2 demethylation catalytic rate k2b
T1
met
hyl
atio
n c
atal
ytic
rat
e k 1
c
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Methylation Catalytic Rate/Demethylation Catalytic Rate = Constant
04/21/23 Yang Yang, Candidacy Seminar 34
T3 demethylation catalytic rate k3b
T2
met
hyl
atio
n c
atal
ytic
rat
e k 2
c
T4 demethylation catalytic rate k4b
T3
met
hyl
atio
n c
atal
ytic
rat
e k 3
c
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Methylation Catalytic Rate/Demethylation Catalytic Rate = Constant
04/21/23 Yang Yang, Candidacy Seminar 35
LT1 demethylation catalytic rate k1bl
LT
0 m
eth
ylat
ion
cat
alyt
ic
rate
k0c
l
LT2 demethylation catalytic rate k2bl
LT
1 m
eth
ylat
ion
cat
alyt
ic
rate
k1c
l
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Methylation Catalytic Rate/Demethylation Catlytic Rate = Constant
04/21/23 Yang Yang, Candidacy Seminar 36
LT3 demethylation catalytic rate k3bl
LT
2 d
emet
hyl
atio
n c
atal
ytic
ra
te k
2cl
LT4 demethylation catalytic rate k4bl
LT
3 d
emet
hyl
atio
n c
atal
ytic
ra
te k
3cl
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Summary: Reaction Kinetics
04/21/23 Yang Yang, Candidacy Seminar 37
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).
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Conditions for Perfect Adaptation:
Protein Concentrations
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Intrinsic Variability in Total Protein Concentrations
04/21/23 Yang Yang, Candidacy Seminar 39
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.
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Relationship Between Protein Concentrations
04/21/23 Yang Yang, Candidacy Seminar 40
(M)
(M)
(M)(M)
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Relationship Between Protein Concentrations (cont’d)
04/21/23 Yang Yang, Candidacy Seminar 41
(M)
(M)
(M)
(M)
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Relationship between Protein Concentrations (cont’d)
04/21/23 Yang Yang, Candidacy Seminar 42
(M)
(M)
(M)
(M)
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Summary: Protein Concentrations
04/21/23 Yang Yang, Candidacy Seminar 43
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).
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Diversity of Chemotaxis Systems
04/21/23 Yang Yang, Candidacy Seminar 44
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
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Example: Two CheY System
04/21/23 Yang Yang, Candidacy Seminar 45
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
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Conclusions
04/21/23 Yang Yang, Candidacy Seminar 46
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).
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Future Extensions
04/21/23 Yang Yang, Candidacy Seminar 47
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).
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Vertebrate Phototransduction
04/21/23 Yang Yang, Candidacy Seminar 48
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
**
***
*
*
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Light Adaptation in Phototransduction
04/21/23 Yang Yang, Candidacy Seminar 49
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)
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Kinetic Model for Vertebrate Phototransduction
04/21/23 Yang Yang, Candidacy Seminar 50
Russell D. Hamer, Visual Neuroscience (2000)
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Mammalian Circadian Clock
04/21/23 Yang Yang, Candidacy Seminar 51
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).
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Conditions in Two-State Receptor Model
04/21/23 Yang Yang, Candidacy Seminar 52
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(
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04/21/23 Yang Yang, Candidacy Seminar 53
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04/21/23 Yang Yang, Candidacy Seminar 54
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04/21/23 Yang Yang, Candidacy Seminar 55
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04/21/23 Yang Yang, Candidacy Seminar 56
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04/21/23 Yang Yang, Candidacy Seminar 57
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04/21/23 Yang Yang, Candidacy Seminar 58
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Checking Dynamics of CheY-P with Solutions
04/21/23 Yang Yang, Candidacy Seminar 59
A
B
C
D
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Protein Concentration Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 60
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Protein Concentration Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 61
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Protein Concentration Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 62
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Protein Concentration Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 63
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Protein Concentration Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 64
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Reaction Rates Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 65
T2 autophosphorylation rate (k2a)
T3 autophosphorylation rate (k3a)
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
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Reaction Rates Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 66
T2 autophosphorylation rate (k2a)
T3 autophosphorylation rate (k3a)
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
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Reaction Rates Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 67
T1 autophosphorylation rate (k1a)
T2 autophosphorylation rate (k2a)
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
Protein concentrations taken from Mello-Tu’s
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Reaction Rates Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 68
T3 autophosphorylation rate (k3a)
T4 autophosphorylation rate (k4a)
inve
rse
of
T3 M
-M c
on
stan
t (K
3B-1)
inve
rse
of
T4
M-M
co
nst
ant
(K4B
-1)
Protein concentrations taken from SPO’s
Protein concentrations taken from Mello-Tu’s
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Reaction Rates Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 69
LT1 autophosphorylation rate (k1al)
LT2 autophosphorylation rate (k2al)
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)
Protein concentrations taken from SPO’s
Protein concentrations taken from Mello-Tu’s
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Reaction Rates Trend Shifting
04/21/23 Yang Yang, Candidacy Seminar 70
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)
Protein concentrations taken from SPO’s
Protein concentrations taken from Mello-Tu’s
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Slices of 3D Surfaces of Parameter Space
04/21/23 Yang Yang, Candidacy Seminar 71
1 2 3 4
5 6 7 8
9 10 111------------------------11
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Slices of 3D Surfaces of Parameter Space
04/21/23 Yang Yang, Candidacy Seminar 72
1 2 3 4
5 6 7 8
9 10 11
1----------------------11
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Slices of 3D Surfaces of Parameter Space Comparing Pair-Wise Relationship
04/21/23 Yang Yang, Candidacy Seminar 73
T1 autophosphorylation rate (k1a)
Inv
ers
e o
f T
1 M
M c
on
sta
nt
(K1
R-1)
T1 autophosphorylation rate (k1a)
Inv
ers
e o
f T
1 M
M c
on
sta
nt
(K1B
-1)
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E. coli and Bacteria Chemotaxis
04/21/23 Yang Yang, Candidacy Seminar 74
http://www.rowland.harvard.edu/labs/bacteria/index_movies.html
Increasing attractants or Decreasing repellents