1. Near-Perfect Adaptation in Bacterial Chemotaxis
Yang Yang and Sima Setayeshgar
Department of Physics
Indiana University, Bloomington, IN
5/10/2019
Yang Yang, Candidacy Seminar

2. Intro slide from SS

3. Intro slide from SS

4. E. coli and Bacteria Chemotaxis
Increasing attractants or Decreasing repellents
5/10/2019
Yang Yang, Candidacy Seminar

5. Chemotaxis Signal Transduction Network in E. coli
Pathway
Motor
Response
[CheY-P]
Stimulus
Flagellar
Bundling
Motion
With approximately 50 interacting proteins , the network converts an external stimulus into an internal stimulus which in turn interacts with the flagella motor to bias the cell’s motion.
CheA: Histidine kinase CheB-P: Methylesterase
CheW: Couples CheA to MCPs CheY-P: Response regulator
CheR: Methyltransferase CheZ: Dephosphorylates CheY-P
It is used as a well-characterized model system for the study of properties of (two-component) cellular signaling networks in general.
Histidine kinase Methylesterase
Couples CheA to MCPs Response regulator
Methyltransferase Dephosphorylates CheY-P
CheB
CheA
CheW
CheZ
CheR
CheY
Run Tumble
5/10/2019
Yang Yang, Candidacy Seminar

6. Robust Perfect Adaptation
From Sourjik et al., PNAS (2002).
Steady state [CheY-P] / running bias
independent of value constant external
stimulus (adaptation)
Precision of adaptation insensitive to changes in network parameters (robustness)
Adaptation Precison
FRET signal [CheY-P]
CheR fold expression
Fast response
Slow adaptation
From Alon et al., Nature (1999).
5/10/2019
Yang Yang, Candidacy Seminar

7. Yang Yang, Candidacy Seminar
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
5/10/2019
Yang Yang, Candidacy Seminar

8. E. coli Chemotaxis Signaling Network
Ligand binding
Methylation
Phosphorylation
phosphorylation
methylation
Ligand binding
Chemotaxis in E. coli involves temporal measurement of the change in concentration of an external stimulus. This is achieved through the existence of fast and slow reaction time scales, in the chemotaxis signal transduction network: fast measurement of the current external concentration is compared with the cell’s “memory” of the concentration some time ago to determine whether to extend a run in a given direction or to tumble, thereby randomly selecting a new direction.
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)
5/10/2019
Yang Yang, Candidacy Seminar

9. Michaelis-Menten Kinetics
Enzymatic reaction:
Where E is the enzyme, 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:
Define E, S, P
Write final rate equation for P in terms of K_m
where Km is the Michaelis Menten Constant (MM constant)
5/10/2019
Yang Yang, Candidacy Seminar

10. Yang Yang, Candidacy Seminar
Reaction Rates
5/10/2019
Yang Yang, Candidacy Seminar

11. Yang Yang, Candidacy Seminar
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.
: state variables
: reaction kinetics
: reaction rates
: external stimulus
5/10/2019
Yang Yang, Candidacy Seminar

12. Yang Yang, Candidacy Seminar
Augmented System
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:
Discretize s in
range {slow, shigh}
5/10/2019
Yang Yang, Candidacy Seminar

13. 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).
: 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)
5/10/2019
Yang Yang, Candidacy Seminar

14. Yang Yang, Candidacy Seminar
Implementation
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:
5/10/2019
Yang Yang, Candidacy Seminar

15. Converting from Guess to Solution
Starting from initial guess A, the solution to B is generated. A
3%<<5%
1%<<3%
0%<<1%
Inverse of T3 MM constant (K3R-1) B
Need large legend for blue, green, red points
T3 autophosphorylation rate (k3a)
5/10/2019
Yang Yang, Candidacy Seminar

16. Parameter Surfaces 0%<<1% 1%<<3% Surface 2D projections
Inverse of T1 methylation MM constant (K1R-1)
Inverse of T1 demethylation MM constant(k1B-1)
Need some plots of *slices*, as opposed to projections (which combine all slices in a given direction)
How do the slices differ from the pair-wise plots where two parameters only are varied?
Inverse of T1 methylation MM constant (K1R-1)
1%<<3%
0%<<1%
T1 autophosphorylation rate K1a
5/10/2019
Yang Yang, Candidacy Seminar

17. Yang Yang, Candidacy Seminar
Validation
Verify steady state NR solutions dynamically using DSODE for different stimulus ramps:
Concentration (µM)
Time (s)
5/10/2019
Yang Yang, Candidacy Seminar

18. Violating and Restoring Perfect Adaptation
Inverse of T3 MM constant (K3R-1)
CheYp Concentration (µM)
T3 autophosphorylation rate (k9)
Time (s)
Step stimulus from 0 to 1e-3M at t=500s
5/10/2019
Yang Yang, Candidacy Seminar

19. Conditions for Perfect Adaptation: Kinetic Parameters
5/10/2019
Yang Yang, Candidacy Seminar

20. Inverse of Methylation MM Constant Autophosphorylation Rate
Inverse of T0 MM constant (K0R-1)
Inverse of T1 MM constant (K1R-1)
T0 autophosphorylation rate (k0a)
T1 autophosphorylation rate (k1a)
5/10/2019
Yang Yang, Candidacy Seminar

21. Inverse of Methylation MM Constant Autophosphorylation Rate
Inverse of T2 MM constant (K2R-1)
Inverse of T3 MM constant (K3R-1)
T2 autophosphorylation rate (k2a)
T3 autophosphorylation rate (k3a)
5/10/2019
Yang Yang, Candidacy Seminar

22. Inverse of Methylation MM Constant Autophosphorylation Rate
Inverse of LT0 MM constant (K0LR-1)
Inverse of LT1 MM constant (K1LR-1)
LT0 autophosphorylation rate (k0al)
LT1 autophosphorylation rate (k1al)
5/10/2019
Yang Yang, Candidacy Seminar

23. Inverse of Methylation MM Constant Autophosphorylation Rate
Inverse of LT2 MM constant (K2LR-1)
Inverse of LT3 MM constant (K3LR-1)
LT2 autophosphorylation rate (k2al)
LT3 autophosphorylation rate (k3al)
5/10/2019
Yang Yang, Candidacy Seminar

24. Inverse of Demethylation MM Constant Autophosphorylation Rate
Inverse of T1 MM constant (K1B-1)
Inverse of T2 MM constant (K2B-1)
T1 autophosphorylation rate (k1a)
T2 autophosphorylation rate (k2a)
5/10/2019
Yang Yang, Candidacy Seminar

25. Inverse of Demethylation MM Constant Autophosphorylation Rate
Inverse of T3 MM constant (K3B-1)
Inverse of T4 MIM constant (K4B-1)
T3 autophosphorylation rate (k3a)
T4 autophosphorylation rate (k4a)
5/10/2019
Yang Yang, Candidacy Seminar

26. Inverse of Demethylation MM Constant Autophosphorylation Rate
Inverse of LT1 MM constant (K1LB-1)
Inverse of LT2 MM constant (K2LB-1)
LT1 autophosphorylation rate (k1al)
LT2 autophosphorylation rate (k2al)
5/10/2019
Yang Yang, Candidacy Seminar

27. Inverse of Demethylation MM Constant Autophosphorylation Rate
Inverse of LT3 MM constant (K2LB-1)
Inverse of LT4 MM constant (K3LB-1)
LT3 autophosphorylation rate (k12)
LT4 autophosphorylation rate (k13)
5/10/2019
Yang Yang, Candidacy Seminar

28. Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant
T1 demethylation catalytic rate
T1 methylation catalytic rate
T2 methylation catalytic rate
T2 demethylation catalytic rate
5/10/2019
Yang Yang, Candidacy Seminar

29. Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant
T3 demethylation catalytic rate
T2 methylation catalytic rate
T3 methylation catalytic rate
T4 demethylation catalytic rate
5/10/2019
Yang Yang, Candidacy Seminar

30. Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant
LT1 demethylation catalytic rate
LT0 methylation catalytic rate
LT1 methylation catalytic rate
LT2 demethylation catalytic rate
5/10/2019
Yang Yang, Candidacy Seminar

31. Methylation Catalytic Rate/ Demethylation Catlytic Rate = Constant
LT3 demethylation catalytic rate
LT2 demethylation catalytic rate
LT3 demethylation catalytic rate
LT4 demethylation catalytic rate
5/10/2019
Yang Yang, Candidacy Seminar

32. Yang Yang, Candidacy Seminar
Summary
The Inverse of Methylation MM constants linearly decrease with Autophosphorylation Rates
The Inverse of Demethylation MM constants linearly increase with Autophosphorylation Rates
The ratio of Methylation catalytic rates and demethylation catlytic rates for the next methylation level is constant for all methylation states
Slides 32 and 33 are redundant
These conditions are consistent with those obtained in previous works from analysis of a detailed, two-state receptor model*.
* B. Mello et al. Biophysical Journal , (2003).
5/10/2019
Yang Yang, Candidacy Seminar

33. 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, whiledissociation 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:
5/10/2019
Yang Yang, Candidacy Seminar

34. Conditions for Perfect Adaptation: Protein Concentrations

35. Summary of Protein Concentrations
Main point is that they are model dependent, with large amount of experimental variability in their measurement where these measurements are available
5/10/2019
Yang Yang, Candidacy Seminar

36. Relationship Between Protein Concentrations
(M)
(M)
(M)
(M)
5/10/2019
Yang Yang, Candidacy Seminar

37. Relationship Between Protein Concentrations (cont’d)
(M)
(M)
(M)
(M)
5/10/2019
Yang Yang, Candidacy Seminar

38. Relationship between Protein Concentrations (cont’d)
(M)
(M)
(M)
(M)
5/10/2019
Yang Yang, Candidacy Seminar

39. Summary For more precise adaptation:
CheR concentration is restricted in a narrow small-value region while total receptor and CheY concentration can vary in a wide region.
CheR concentration is proportional to the CheB concentration
For smaller or larger value of CheY concentration, total receptor concentration can vary in a wide region.
CheB concentration is restricted in a narrow region while total receptor and CheY concentration can vary in a wide region.
Concluding remarks for this part

40. Diversity of Chemotaxis Systems
In different bacteria, additional protein components as well as multiple copies of certain chemotaxis proteins are present.
Response regulator
Phosphate “sink”
CheY1
CheY2
Eg., Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteria possess multiple CheYs while lacking of CheZ homologue.
5/10/2019
Yang Yang, Candidacy Seminar

41. Yang Yang, Candidacy Seminar
Two CheY System
Exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ:
CheY1p (µM)
Time(s)
Time(s)
Requiring:
Faster phosphorylation/autodephosphorylation rates of CheY2
than CheY1
Faster phosphorylation rate of CheB
5/10/2019
Yang Yang, Candidacy Seminar

42. Yang Yang, Candidacy Seminar
Conclusions
Successful implementation of a novel method for elucidating regions in parameter space allowing precise adaptation
Numerical results for (near-) perfect adaptation manifolds in parameter space for the E. coli chemotaxis network, allowing determination of
Conditions required for perfect adaptation, consistent with and extending previous works [1-3]
Numerical ranges for experimentally unknown or partially known kinetic parameters
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).
5/10/2019
Yang Yang, Candidacy Seminar

43. Yang Yang, Candidacy Seminar
Future Work
Extension to other signaling networks
vertebrate phototransduction
mammalian circadian clock
allowing determination of
parameter dependences underlying robustness of adaptation
b) plausible numerical values for unknown network parameters
5/10/2019
Yang Yang, Candidacy Seminar

44. Vertebrate Phototransduction
cGMP: cyclic GMP
PDE: cGMP phosphodiesterase
GCAP: guanylyl cyclase activating, Ca2+ binding protein
gc: guanylyl cyclase, which synthesis cGMP
5/10/2019
Yang Yang, Candidacy Seminar

45. Light Adaptation of 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. (Neuroscience, Purves et al., 2001)
5/10/2019
Yang Yang, Candidacy Seminar

46. Kinetic Model for Vertebrate Phototransduction
Russell D. Hamer, Visual Neuroscience (2000)
5/10/2019
Yang Yang, Candidacy Seminar

47. Mammalian Circadian Clock
From Forger et al., PNAS (2003).
PERs transport CRYs to nucleus
CLOCK and BMAL1 bind together
CLOCK·BMAL1 binds to E box to increase Pers(Crys) transcription rates
E box is the sequence CACGTG of the PER1 and CRY1 genes
PERs bind with kinases CKIε/δ to be phosphorylated
Phosphorylated PERs bind with CRYs
Only phosphorylated PER·CRY· CKIε/δ can enter nucleus
Phosphorylated PER·CRY· CKIε/δ inhibit the ability of CLOCK·BMALI to enhance transcription
Increasing REV-ERBα levels repress BMAL1 transcription
Activator positively regulated BMAL1 transcription
5/10/2019
Yang Yang, Candidacy Seminar

48. Yang Yang, Candidacy Seminar
5/10/2019
Yang Yang, Candidacy Seminar

49. Yang Yang, Candidacy Seminar
5/10/2019
Yang Yang, Candidacy Seminar

50. Yang Yang, Candidacy Seminar
5/10/2019
Yang Yang, Candidacy Seminar

51. Yang Yang, Candidacy Seminar
5/10/2019
Yang Yang, Candidacy Seminar

52. Yang Yang, Candidacy Seminar
5/10/2019
Yang Yang, Candidacy Seminar

53. Yang Yang, Candidacy Seminar
5/10/2019
Yang Yang, Candidacy Seminar

54. Checking Dynamics of CheY-P with Solutions
A B C D
5/10/2019
Yang Yang, Candidacy Seminar

55. Protein Concentration Trend Shifting
5/10/2019
Yang Yang, Candidacy Seminar

56. Protein Concentration Trend Shifting
5/10/2019
Yang Yang, Candidacy Seminar

57. Protein Concentration Trend Shifting
5/10/2019
Yang Yang, Candidacy Seminar

58. Protein Concentration Trend Shifting
5/10/2019
Yang Yang, Candidacy Seminar

59. Protein Concentration Trend Shifting
5/10/2019
Yang Yang, Candidacy Seminar

60. Reaction Rates Trend Shifting
Protein concentrations taken from SPO’s
inverse of T2 MM constant (K2R-1)
inverse of T3 MM constant (K3R-1)
Protein concentrations taken from Mello-Tu’s
T2 autophosphorylation rate (k2a)
T3 autophosphorylation rate (k3a)
5/10/2019
Yang Yang, Candidacy Seminar

61. Reaction Rates Trend Shifting
Protein concentrations taken from SPO’s
inverse of T2 MM constant (K2R-1)
inverse of T3 MM constant (K3R-1)
Protein concentrations taken from Mello-Tu’s
T2 autophosphorylation rate (k2a)
T3 autophosphorylation rate (k3a)
5/10/2019
Yang Yang, Candidacy Seminar

62. Reaction Rates Trend Shifting
Protein concentrations taken from SPO’s
inverse of T1 M-M constant (K1B-1)
inverse of T2 M-M constant (K2B-1)
Protein concentrations taken from Mello-Tu’s
T1 autophosphorylation rate (k1a)
T2 autophosphorylation rate (k2a)
5/10/2019
Yang Yang, Candidacy Seminar

63. Reaction Rates Trend Shifting
Protein concentrations taken from SPO’s
inverse of T3 M-M constant (K3B-1)
inverse of T4 M-M constant (K4B-1)
Protein concentrations taken from Mello-Tu’s
T3 autophosphorylation rate (k3a)
T4 autophosphorylation rate (k4a)
5/10/2019
Yang Yang, Candidacy Seminar

64. Reaction Rates Trend Shifting
Protein concentrations taken from SPO’s
inverse of LT1 MM constant (K1LB-1)
inverse of LT2 MM constant (K2LB-1)
Protein concentrations taken from Mello-Tu’s
LT1 autophosphorylation rate (k1al)
LT2 autophosphorylation rate (k2al)
5/10/2019
Yang Yang, Candidacy Seminar

65. Reaction Rates Trend Shifting
Protein concentrations taken from SPO’s
inverse of LT3 MM constant (K2LB-1)
inverse of LT4 MM constant (K3LB-1)
Protein concentrations taken from Mello-Tu’s
LT3 autophosphorylation rate (k12)
LT4 autophosphorylation rate (k13)
5/10/2019
Yang Yang, Candidacy Seminar

66. Slices of 3D Surfaces of Parameter Space
5/10/2019
Yang Yang, Candidacy Seminar

67. Slices of 3D Surfaces of Parameter Space
5/10/2019
Yang Yang, Candidacy Seminar