1. Near-Perfect Adaptation in Bacterial Chemotaxis
Yang Yang and Sima Setayeshgar
Department of Physics
Indiana University, Bloomington, IN
11/29/2018
Yang Yang, Candidacy Seminar

2. 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.
E. coli is a flagellated bacterium, about 1 micron in length and 0.5 micron in diameter, with a life cycle of approx. 1 hour. It has been the work horse of molecular biology for the past 70 years, due its usual lack of pathogenicity (the dangerous strains are actually pretty rare), ease of growth in the lab, and relatively small genome. It has developed as a basis for recent developments in biotechnology and genetic engineering and for understanding fundamental cellular processes -- cellular sensory systems (how cells sense and respond to their environments), regulation of gene expression, cell division, etc. – where the goal is to look at these processes in the well-controllable and experimentally accessible context of E. coli, and learn not just how they work but what the design principles are (why they work the way do), thereby allowing us to extend what we learn to other organisms, such as ourselves.
Dimensions:
Body size: 1 μm in length
0.4 μm in radius
Flagellum: 10 μm long
Cell cycle:
~ 1 hour
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3. Yang Yang, Candidacy Seminar
E. coli in Motion
From Berg & Brown, Nature (1972).
Increasing attractants or Decreasing repellents
Chemotaxis in bacteria, such as E-Coli, which is a biased random walk of "runs" punctuated by "tumbles": Shown on the left is a 3-d trajectory of
a single bacterium. As seen in the movie, each bacterium has 6-8 flagella, with an inherent chirality: CCW rotation of the motor apparatus at the base of each flagellum allows the flagella to bundle up and act as a single propeller, leading to running motion of the cell. CW rotation leads to flying apart of the bundle, resulting in tumbling behavior that allows a new, perhaps more favorable direction of motion to be randomly selected. In the top right, the cell speed is shown as a function, indicating smooth running periods at constant speed, followed by stationary tumbles.
Temporal measurement of the external concentration, and comparison with the cell's memory of it some time ago, are used to modulate the mean runtime. “Memory” in E. coli is achieved through the existence of fast and slow time scales in the biochemical signaling cascade that governs the chemotactic response.
Physical constants:
Cell speed: μm/sec
Mean run time: 1 sec
Mean tumble time: 0.1 sec
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4. Bacterial Chemotaxis
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.
“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 E. coli chemotaxis signal transduction network has emerged as a canonical example of cellular sensory systems: all chemotaxis proteins are known, most have been crystallized, many reaction rate constants are well-characterized. As such, it is an excellent testbed for understanding the design principles giving rise to important properties of many signal transduction networks, namely adaptation to the DC level of input (which for E. coli takes place over 10^4 – 10^5 orders of magnitude of input concentration), large amplification in the input-output response (which for E. coli is approximately 50), and robustness to variations in reaction rates.

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 – the change in concentration of the phosphorylated form of the Y chemotaxis protein, CheY - 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
The existence of fast and slow time scales in the biochemical signaling cascade leading from signal detection to motor response is achieved through the existence of fast reaction steps (ligand binding, phosphorylation) and slow reaction steps (methylation, demethylation).
Histidine kinase Methylesterase
Couples CheA to MCPs Response regulator
Methyltransferase Dephosphorylates CheY-P
CheB
CheA
CheW
CheZ
CheR
CheY
Run Tumble
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6. Flagellar Motor in E. coli
In addition to the signal transduction network, the rotational bias and switching frequency of a single flagellum as a function of the concentration of the intracellular response regulator CheY-P have been quantitatively measured, as shown on the right. These are experiments are done on tethered cells, as shown on the left.
From R. M. Berry, Encyclopedia of Life Science (2001).
From P. Cluzel, et al., Science (2000).
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7. Response to Step Stimulus
Flagellar response:
CheY-P response:
From Block et al., Cell (1982).
The flagellar response to a step stimulus is shown on the left. Recent experiments are able to probe the dynamics of intracellular protein concentrations governing this response, for example using FRET (fluorescence resonance energy transfer), as shown on the right. These experiments show fast change in the CheY-P concentration in response to a step stimulus, followed by slow adaptation to the prestimulus value.
Steady state [CheY-P] ( and running bias)
independent of value constant external
stimulus (adaptation)
Fast response
Slow adaptation
From Sourjik et al., PNAS (2002).
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8. Excitation and Adaptation
This fast response and slow recovery is depicted schematically here, demonstrating that adaptation in the chemotaxis network is achieved through a balance between the fast and slow reaction kinetics.
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9. Precision of Adaptation
Squares: Unstimulated cells
Circles: Cells stimulated at t=0
(Each point represents data from 10s motion of cells.)
Precision of adaptation =
steady state tumbling
frequency of unstimulated cells /
frequency of stimulated cells
Experimentally, it is found that adaption in E. coli is precise: That is, the pre- and post-stimulus values of the internal response regulator CheY are very nearly equal.
From Alon et al. Nature (1999).
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10. Robustness of Perfect Adaptation
Precision of adaptation robust to 50-fold change in CheR expression …
…while …
Adaptation time and steady state tumbling frequency vary significantly.
Furthermore, this property of the chemotaxis network is “robust” to variations in the network parameters, such as reaction rates and concentrations of proteins catalyzing various reactions. On the left, experimental results showing variation in the precision of adaptation as a function of one of these protein concentrations is shown. It should be pointed while this property of the network – the precision of adaptation – is “robust”, others such as the network adapation time are not.
Robustness of perfect adaptation: precision of adaptation insenstive to network parameters
From Alon et al. Nature (1999).
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11. 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
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12. 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)
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13. Michaelis-Menten Kinetics
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:
where Km is the Michaelis-Menten (MM) constant
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14. Yang Yang, Candidacy Seminar
Reaction Rates
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15. 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
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16. 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}
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17. 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)
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18. 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:
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19. Convergence from Guess to Solution
Starting from initial guess A, solution B is generated. By comprehensively sampling space of parameters with initial guesses, solution “surfaces” are constructed.
3%<<5%
1%<<3%
0%<<1% A
Inverse of T3 MM constant (K3R-1) B
Need large legend for blue, green, red points
Make relationship with delta and epsilon
T3 autophosphorylation rate (k3a)
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20. 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
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21. Slices of 3D Surfaces of Parameter Space
1 -11 denote slices
perpendicular to K1B-1
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22. Yang Yang, Candidacy Seminar
Validation
Verify steady state NR solutions dynamically using DSODE for different stimulus ramps:
Concentration (µM)
Time (s)
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23. Violating and Restoring Perfect Adaptation
Inverse of T3 MM constant (K3R-1)
CheYp Concentration (µM)
T3 autophosphorylation rate (k3a)
Time (s)
Step stimulus from 0 to 1e-3M at t=500s
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24. Conditions for Perfect Adaptation: Kinetic Parameters
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25. 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)
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26. 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)
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27. 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)
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28. 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)
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29. 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)
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30. 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)
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31. 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)
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32. Inverse of Demethylation MM Constant Autophosphorylation Rate
Inverse of LT3 MM constant (K2LB-1)
Inverse of LT4 MM constant (K3LB-1)
LT3 autophosphorylation rate (k3al)
LT4 autophosphorylation rate (k4al)
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33. Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant
T1 demethylation catalytic rate k1b
T0 methylation catalytic rate k0c
T1 methylation catalytic rate k1c
T2 demethylation catalytic rate k2b
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34. Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant
T3 demethylation catalytic rate k3b
T2 methylation catalytic rate k2c
T3 methylation catalytic rate k3c
T4 demethylation catalytic rate k4b
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35. Methylation Catalytic Rate/ Demethylation Catalytic Rate = Constant
LT1 demethylation catalytic rate k1bl
LT0 methylation catalytic rate k0cl
LT1 methylation catalytic rate k1cl
LT2 demethylation catalytic rate k2bl
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36. Methylation Catalytic Rate/ Demethylation Catlytic Rate = Constant
LT3 demethylation catalytic rate k3bl
LT2 demethylation catalytic rate k2cl
LT3 demethylation catalytic rate k3cl
LT4 demethylation catalytic rate k4bl
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37. Summary: Reaction Kinetics
Inverse of methylation MM constants linearly decreases with autophosphorylation rates
Inverse of demethylation MM constants linearly increases with autophosphorylation rates
Ratio of methylation catalytic rates and demethylation catalytic 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 (activity-based) receptor model*.
* B. Mello et al. Biophysical Journal (2003).
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38. Conditions for Perfect Adaptation: Protein Concentrations

39. 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.
Main point is that they are model dependent, with large amount of experimental variability in their measurement where these measurements are available
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40. Relationship Between Protein Concentrations
(M)
(M)
(M)
(M)
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41. Relationship Between Protein Concentrations (cont’d)
(M)
(M)
(M)
(M)
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42. Relationship between Protein Concentrations (cont’d)
(M)
(M)
(M)
(M)
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43. 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).
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44. 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.
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45. Example: 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
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46. 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
Preliminary results on restrictions of total protein concentrations
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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47. Yang Yang, Candidacy Seminar
Future Extensions
Extension to other signaling networks
- vertebrate phototransduction [1]
- mammalian circadian clock [2]
allowing determination of
parameter dependences underlying robustness of adaptation
b) plausible numerical values for unknown network parameters
[1] R. D. Hamer, et al., Vis Neurosci 22, (2005).
[2] D. B. Forger and C. S. Peskin, PNAS 102, (2005).
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48. Vertebrate Phototransduction
cGMP: cyclic GMP
PDE: cGMP phosphodiesterase
GCAP: guanylyl cyclase
gc: guanylyl cyclase
From
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49. 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)
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50. Kinetic Model for Vertebrate Phototransduction
Russell D. Hamer, Visual Neuroscience (2000)
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51. 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
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52. 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:
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53. Yang Yang, Candidacy Seminar
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54. Yang Yang, Candidacy Seminar
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55. Yang Yang, Candidacy Seminar
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56. Yang Yang, Candidacy Seminar
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57. Yang Yang, Candidacy Seminar
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58. Yang Yang, Candidacy Seminar
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59. Checking Dynamics of CheY-P with Solutions
A B C D
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60. Protein Concentration Trend Shifting
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61. Protein Concentration Trend Shifting
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62. Protein Concentration Trend Shifting
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63. Protein Concentration Trend Shifting
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64. Protein Concentration Trend Shifting
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65. 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)
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66. 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)
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67. 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)
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68. 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)
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69. 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)
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70. 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)
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71. Slices of 3D Surfaces of Parameter Space
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72. Slices of 3D Surfaces of Parameter Space
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73. T1 autophosphorylation rate (k1a) T1 autophosphorylation rate (k1a)
Slices of 3D Surfaces of Parameter Space Comparing Pair-Wise Relationship
Inverse of T1 MM constant (K1B-1)
Inverse of T1 MM constant (K1R-1)
T1 autophosphorylation rate (k1a)
T1 autophosphorylation rate (k1a)
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74. E. coli and Bacteria Chemotaxis
Increasing attractants or Decreasing repellents
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