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Calcium
1. Invented by Nature, Rediscovered by Man:
Feedback Control Systems in Biology and Engineering
Mustafa Khammash
University of California, Santa Barbara
4. Outline
❖ Feedback control at the system level
‣ Calcium homeostasis in mammals
❖ Feedback control at the molecular level
‣ The bacterial Heat-Shock Response
5. Outline
❖ Feedback control at the system level
‣ Calcium homeostasis in mammals
❖ Feedback control at the molecular level
‣ The bacterial Heat-Shock Response
❖ A Bio-Inspired Engineering Application
‣ Search strategies for unmanned aerial vehicles
6. Outline
❖ Feedback control at the system level
‣ Calcium homeostasis in mammals
❖ Feedback control at the molecular level
‣ The bacterial Heat-Shock Response
❖ A Bio-Inspired Engineering Application
‣ Search strategies for unmanned aerial vehicles
❖ Challenges and opportunities
7. Outline
❖ Feedback control at the system level
‣ Calcium homeostasis in mammals
❖ Feedback control at the molecular level
‣ The bacterial Heat-Shock Response
❖ A Bio-Inspired Engineering Application
‣ Search strategies for unmanned aerial vehicles
❖ Challenges and opportunities
❖ Conclusions
11. Homeostasis
❖ "Homeostasis" is derived from the Greek words for "same" and
"steady."
❖ Refers to ways the body acts to maintain a stable internal
environment.
4
12. Homeostasis
❖ "Homeostasis" is derived from the Greek words for "same" and
"steady."
❖ Refers to ways the body acts to maintain a stable internal
environment.
❖ The body is endowed with a multitude of automatic mechanisms of
feedback that counteract influences tending toward disequilibrium.
4
13. Homeostasis
❖ "Homeostasis" is derived from the Greek words for "same" and
"steady."
❖ Refers to ways the body acts to maintain a stable internal
environment.
❖ The body is endowed with a multitude of automatic mechanisms of
feedback that counteract influences tending toward disequilibrium.
❖ In history:
‣ Claude Bernard (1865) -- Fixité du milieu intérieur
‣ Walter Cannon (1929) -- Homeostasis
‣ Norbert Wiener (1948) -- Cybernetics
4
14. Physiological Role of Calcium
❖ Maintain the integrity of the skeleton.
❖ Control of biochemical processes:
‣ Intracellular:
- Activity of a large number of enzymes
- Conveying information from the surface to the interior of the cell
‣ Extracellular:
- Muscle and nerve function
- Blood clotting
16. Calcium Homeostasis in Mammals
❖ The biochemical role of Calcium requires that its blood plasma
concentrations be precisely controlled
17. Calcium Homeostasis in Mammals
❖ The biochemical role of Calcium requires that its blood plasma
concentrations be precisely controlled
❖ Normal concentration of about 9 mg/dl must be maintained within
small tolerances despite
‣ variations in dietary calcium levels
‣ variation in demand for calcium
18. Calcium Homeostasis in Mammals
❖ The biochemical role of Calcium requires that its blood plasma
concentrations be precisely controlled
❖ Normal concentration of about 9 mg/dl must be maintained within
small tolerances despite
‣ variations in dietary calcium levels
‣ variation in demand for calcium
❖ Humans and other mammals have an effective feedback mechanism
for regulating plasma concentration of calcium [Ca]p
20. Calcium Homeostasis in Dairy Cows
Ca Clearance Rate
100
90
❖ Plasma concentrations are 80
70
easily maintained during g/day
60
periods of nonlactation 50
40
30
20
10
0
10 12 14 16 18 20 22
time (days)
Plasma Ca Concentration
0.1
0.095
0.09
0.085
g/l 0.08
0.075
0.07
0.065
0.06
0.055
0.05
10 12 14 16 18 20 22
time (days)
Parturition
21. Calcium Homeostasis in Dairy Cows
Ca Clearance Rate
100
90
❖ Plasma concentrations are 80
70
easily maintained during g/day
60
periods of nonlactation 50
40
30
❖ An especially large loss of 20
plasma calcium to milk takes
10
0
10 12 14 16 18 20 22
place during lactation time (days)
Plasma Ca Concentration
0.1
0.095
0.09
0.085
g/l 0.08
0.075
0.07
0.065
0.06
0.055
0.05
10 12 14 16 18 20 22
time (days)
Parturition
22. Calcium Homeostasis in Dairy Cows
Ca Clearance Rate
100
90
❖ Plasma concentrations are 80
70
easily maintained during g/day
60
periods of nonlactation 50
40
30
❖ An especially large loss of 20
plasma calcium to milk takes
10
0
10 12 14 16 18 20 22
place during lactation time (days)
Plasma Ca Concentration
❖ Most animals adapt to the 0.1
0.095
onset of lactation 0.09
0.085
g/l 0.08
0.075
0.07
0.065
0.06
0.055
0.05
10 12 14 16 18 20 22
time (days)
Parturition
23. A Disorder of Calcium Homeostasis
❖ In some animals, the regulatory
mechanism fails to meet the
increased calcium demand
❖ Animals become hypocalcemic
‣ Results in disruption of muscle and
nerve function
‣ Leads to recumbency
❖ The clinical syndrome is Parturient
Paresis (Milk Fever)
❖ Affects 6% of the dairy cows in the
US
24. Calcium Flow
Milk, fetus
Formation Filtration
Bone Calcium pool Kidney
Resorption reabsorption
Secretion Absorption
Intestine
26. Mathematical Modeling of [Ca]
Ca Total Supply Rate
VT (g/day)
Intestinal Absorption
Plasma
Bone Resorption
27. Mathematical Modeling of [Ca]
Ca Total Supply Rate Total Ca Clearance Rate
VT (g/day) Vcl (g/day)
Intestinal Absorption
Plasma Milk, fetus, urine, etc.
Bone Resorption
28. Mathematical Modeling of [Ca]
Ca Total Supply Rate Total Ca Clearance Rate
VT (g/day) Vcl (g/day)
Intestinal Absorption
Plasma Milk, fetus, urine, etc.
Bone Resorption
Vol = Plasma Volume (l)
[Ca]p = Plasma Concentration (g/l)
29. in block diagram form...
1 t
[Ca]p = (VT − Vcl )dτ
Vol 0
Vcl
-
VT + k
30. Vcl
Set point e VT -
+
+ Control
-
e = error (g/l)
what is f (·) ?
32. Standard Model
❖ A model describing the relation between VT and [Ca]p is given by:
Source: Ramberg, Johnson, Fargo, and Kronfeld, “Calcium homeostasis in cows, with special reference to
parturient hypocalcemia,” Am. J. Physiol. , 1984.
33. Standard Model
❖ A model describing the relation between VT and [Ca]p is given by:
Source: Ramberg, Johnson, Fargo, and Kronfeld, “Calcium homeostasis in cows, with special reference to
parturient hypocalcemia,” Am. J. Physiol. , 1984.
This is proportional feedback! VT = Kp e
34. Deficiencies in the Standard Model
❖ From basic principles of control theory, proportional feedback alone
cannot explain:
‣ The observed zero steady-state error
(Perfect Adaptation)
‣ The shape of the time response of [Ca]p following increased Calcium
clearance at calving
35. Integral Feedback
❖ In order to account for the zero state-state error integral feedback must
be present.
❖ When combined with Proportional Feedback, Integral Feedback will
account for
‣ The zero steady-state error in response to Ca clearance
‣ The second order shape of the [Ca]p time response
❖ We propose the feedback:
36. Implications of PI Feedback
PI Feedback
Vcl
Set point
e
VT
+
-
+ +
-
❖ Supply rate depends on both the level and duration of calcium
deficiency prior to and until the time of interest.
❖ Understanding the dynamics of the system is unavoidable.
37. Model vs. Experiment
❖ Data from two groups of
normal lactating dairy
cows around the day of
calving (NADC)
❖ One group was used to
determine model
parameters
❖ The model prediction
was compared against
data from the second
group (20 animals)
17
39. How Is Integral Action Realized?
❖ Our model was arrived at through necessity arguments
40. How Is Integral Action Realized?
❖ Our model was arrived at through necessity arguments
❖ Is there a plausible physiological basis?
41. How Is Integral Action Realized?
❖ Our model was arrived at through necessity arguments
❖ Is there a plausible physiological basis?
❖ Given that calcium is hormonally regulated, what is the mechanism
through which integration is realized?
42. How Is Integral Action Realized?
❖ Our model was arrived at through necessity arguments
❖ Is there a plausible physiological basis?
❖ Given that calcium is hormonally regulated, what is the mechanism
through which integration is realized?
Can a single hormone be at work?
• P feedback:
• PI feedback:
54. Hormonal Regulation
The Parathyroid Gland monitors blood
calcium and secretes Parathyroid Hormone
(PTH) in proportion to [Ca] deficiency
PTH stimulates renal calcium
reabsorption and bone resorption
55. Hormonal Regulation
The Parathyroid Gland monitors blood
calcium and secretes Parathyroid Hormone
(PTH) in proportion to [Ca] deficiency
PTH stimulates renal calcium
reabsorption and bone resorption
56. Hormonal Regulation
The Parathyroid Gland monitors blood
calcium and secretes Parathyroid Hormone
(PTH) in proportion to [Ca] deficiency
PTH stimulates renal calcium
reabsorption and bone resorption
(1,25 OH2 D3) Hormone stimulates
calcium absorption from the intestine
57. Hormonal Regulation
The Parathyroid Gland monitors blood
calcium and secretes Parathyroid Hormone
(PTH) in proportion to [Ca] deficiency
PTH stimulates renal calcium
reabsorption and bone resorption
(1,25 OH2 D3) Hormone stimulates
calcium absorption from the intestine
58. Hormonal Regulation
The Parathyroid Gland monitors blood
calcium and secretes Parathyroid Hormone
(PTH) in proportion to [Ca] deficiency
PTH stimulates renal calcium
reabsorption and bone resorption
(1,25 OH2 D3) Hormone stimulates
calcium absorption from the intestine
Bone resporption and intestinal absorption
account for the entire calcium supply
59. Hormonal Regulation
The Parathyroid Gland monitors blood
calcium and secretes Parathyroid Hormone
(PTH) in proportion to [Ca] deficiency
PTH stimulates renal calcium
reabsorption and bone resorption
(1,25 OH2 D3) Hormone stimulates
calcium absorption from the intestine
Bone resporption and intestinal absorption
account for the entire calcium supply
60. The Integral Term
• Two forms of Vitamin D: 25 (OH)D and 1,25 (OH)2 D
• PTH activates 25 (OH)D in the kidney to form 1,25 OH2 D
PTH
25 (OH)D 1,25 (OH)2D
For a given [25 (OH)D]:
62. Understanding Milk Fever
❖ The supply of calcium from the bone cannot be increased
indefinitely in response to an increases in [PTH]
❖ Absorption is transiently reduced as a result of low calcium
Vcl
Set point e
VT -
+
+ +
-
x
63. Breakdown Is Observed in Nonlinear Model
Phase Portrait for Kp=3000, Ki=1200 Phase Portrait for Kp=5000, Ki=3000
Initial condition
(low clearance EP)
Breakdown Homeostasis is achieved
64. Summary
❖ Calcium homeostasis is achieved through integral feedback. Integral
action is realized by the dynamic interaction among 1,25 (OH)2D and
PTH
❖ Sequence of discovery: Perfect adaptation necessity of integral
action specific action at molecular level
❖ The dynamic interactions give a new perspective on calcium
homeostasis disorders and disease trajectories
❖ Future work:
‣ Other homeostatic mechanisms, e.g. blood sugar, diabetes
‣ Osteoporosis
26
65. Control at the Molecular Level
Bacterial Heat Shock Response
69. Gene Expression
Translation proteins
mRNA
Transcription ribosomes
mRNA
start
DNA
mRNA
end
promoter
RNA polymerase DNA
70. Gene Expression
Translation proteins
mRNA
Transcription ribosomes
mRNA
protein
start
DNA
mRNA
end
promoter
RNA polymerase DNA
71. Gene Expression
Translation proteins
Central Dogma of
Molecular Biology
mRNA
Transcription ribosomes
mRNA
protein
start
DNA
mRNA
end
promoter
RNA polymerase DNA
82. The Heat-Shock Response
❖ High temperatures lead to heat induced stress due to a large
increase in protein unfolding/misfolding
83. The Heat-Shock Response
❖ High temperatures lead to heat induced stress due to a large
increase in protein unfolding/misfolding
❖ The heat-shock response is a protective cellular response to deal
with heat-induced protein damage.
84. The Heat-Shock Response
❖ High temperatures lead to heat induced stress due to a large
increase in protein unfolding/misfolding
❖ The heat-shock response is a protective cellular response to deal
with heat-induced protein damage.
❖ It involves building and dispatching heat-shock proteins (HSPs)
‣ Chaperones: refold denatured proteins
‣ Proteases: degrade aggregated proteins
86. Function of the Heat-Shock Proteins
I. Protein Folding
DnaK/J GroEL/
GroES
Unfolded/partially folded
Proteins
Folded
Proteins
87. Function of the Heat-Shock Proteins
I. Protein Folding
DnaK/J GroEL/
GroES
Unfolded/partially folded
Proteins
II. Protein Degradation
Proteases
Folded
Amino Acids Proteins
Proteins Aggregates
101. II. Regulation of σ32 Activity: A Feedback Mechanism
RNAP hsp1 hsp2
102. II. Regulation of σ32 Activity: A Feedback Mechanism
RNAP hsp1 hsp2
Transcription & Translation
Chaperones
103. II. Regulation of σ32 Activity: A Feedback Mechanism
RNAP hsp1 hsp2
Transcription & Translation
Chaperones
Heat
104. II. Regulation of σ32 Activity: A Feedback Mechanism
RNAP hsp1 hsp2
Transcription & Translation
Chaperones
Heat
105. II. Regulation of σ32 Activity: A Feedback Mechanism
RNAP hsp1 hsp2
Transcription & Translation
Chaperones
Heat
106. III. Regulation of σ32 Degradation
FtsH degrades sigma-32 only when bound to chaperones
RNAP hsp1 hsp2
Transcription & Translation
Chaperones
Heat
107. III. Regulation of σ32 Degradation
FtsH degrades sigma-32 only when bound to chaperones
RNAP hsp1 hsp2
Transcription & Translation
Proteases Chaperones
Heat
FtsH
FtsH
FtsH
108. III. Regulation of σ32 Degradation
FtsH degrades sigma-32 only when bound to chaperones
RNAP hsp1 hsp2
Transcription & Translation
Proteases Chaperones
Heat
FtsH
FtsH
FtsH
FtsH
109. III. Regulation of σ32 Degradation
FtsH degrades sigma-32 only when bound to chaperones
RNAP hsp1 hsp2
Transcription & Translation
Proteases Chaperones
Heat
FtsH
FtsH
FtsH
FtsH
110. Disturbance
FF sensor
FB
Control sensor
Plant
Actuator
A control theorist’s view.
What is the relation to the HS system?
114. 600 24000
DnaK
450
Total σ32 16000
16000
300
12000
150
0 8000
0 10 20
0 10 20
0
Time (min)
30 42o 30 42o
o o
8 E+ 06
6
4
Free σ32 Unfolded Proteins
2
0 0
0 10 20
0 0 10 20
Time (min) Time (min)
115. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
‣ Disturbance feedforward
‣ Activity feedback loop
‣ Degradation feedback loop
‣ High sigma-32 flux
❖ What lies behind the complexity?
116. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
‣ Disturbance feedforward
‣ Activity feedback loop
‣ Degradation feedback loop
‣ High sigma-32 flux
❖ What lies behind the complexity?
Analysis Tools
117. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
‣ Disturbance feedforward
‣ Activity feedback loop
‣ Degradation feedback loop
‣ High sigma-32 flux
❖ What lies behind the complexity?
Analysis Tools
‣ Dynamic analysis and simulations
118. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
‣ Disturbance feedforward
‣ Activity feedback loop
‣ Degradation feedback loop
‣ High sigma-32 flux
❖ What lies behind the complexity?
Analysis Tools
‣ Dynamic analysis and simulations
‣ Sensitivity/robustness analysis
119. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
‣ Disturbance feedforward
‣ Activity feedback loop
‣ Degradation feedback loop
‣ High sigma-32 flux
❖ What lies behind the complexity?
Analysis Tools
‣ Dynamic analysis and simulations
‣ Sensitivity/robustness analysis
‣ Sum-of-Squares tools
120. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
‣ Disturbance feedforward
‣ Activity feedback loop
‣ Degradation feedback loop
‣ High sigma-32 flux
❖ What lies behind the complexity?
Analysis Tools
‣ Dynamic analysis and simulations
‣ Sensitivity/robustness analysis
‣ Sum-of-Squares tools
‣ Optimal control
121. 600
450
Total σ32
300
150 Wild type
0
0 10 20
0
Time (min)
30o 42o
122. 600
450
Total σ32
300
150 Wild type
No feedforward
0
0 10 20
0
Time (min)
30o 42o
123. 600
450
Total σ32
300
150 Wild type
No feedforward
0
0 10 20
0 No DnaK interaction
Time (min)
30o 42o
124. 600
450
Total σ32
300
150 Wild type
No feedforward
0
0 10 20
0 No DnaK interaction
Time (min)
Constitutive σ32 degradation
30o 42o
125. 600
450
Total σ32
300
150 Wild type
No feedforward
0
0 10 20
0 No DnaK interaction
Time (min)
Constitutive σ32 degradation
30o 42o Low σ32 flux
126. 600 2400
DnaK
450
Total σ32 1600
300 1600
1200
150 Wild type
No feedforward
800
0
0 10 20
0 10 No DnaK interaction
20
0
Time (min)
Constitutive σ32 degradation
30 42o Low σ32 flux
o
127. 600 2400
DnaK
450
Total σ32 1600
300 1600
1200
150 Wild type
No feedforward
800
0
0 10 20
0 10 No DnaK interaction
20
0
Time (min)
Constitutive σ32 degradation
30 42o Low σ32 flux
o
8
6
4
Free σ32
2
0 10 20
0
0
Time (min)
128. 600 2400
DnaK
450
Total σ32 1600
300 1600
1200
150 Wild type
No feedforward
800
0
0 10 20
0 10 No DnaK interaction
20
0
Time (min)
Constitutive σ32 degradation
30 42o Low σ32 flux
o
8 E+ 06
6
4
Free σ32 Unfolded Proteins
2
0 0
0 10 20
0 0 10 20
Time (min) Time (min)
129. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
130. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
– Disturbance feedforward
131. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
– Disturbance feedforward Fast response
132. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
– Disturbance feedforward Fast response
– Activity feedback loop
133. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
– Disturbance feedforward Fast response
– Activity feedback loop Robustness, efficiency
134. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
– Disturbance feedforward Fast response
– Activity feedback loop Robustness, efficiency
– Degradation feedback loop
135. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
– Disturbance feedforward Fast response
– Activity feedback loop Robustness, efficiency
– Degradation feedback loop Fast response, noise suppression
136. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
– Disturbance feedforward Fast response
– Activity feedback loop Robustness, efficiency
– Degradation feedback loop Fast response, noise suppression
– High sigma-32 flux
137. Analysis of the Heat-Shock System
❖ What are the advantages of the different control strategies
– Disturbance feedforward Fast response
– Activity feedback loop Robustness, efficiency
– Degradation feedback loop Fast response, noise suppression
– High sigma-32 flux Fast response
139. Is the Wild Type Optimal?
❖ Observation: A hypothetical heat-shock response system maybe
devised to achieve
‣ a very small number of unfolded proteins
‣ minimal complexity (no feedback necessary)
140. Is the Wild Type Optimal?
❖ Observation: A hypothetical heat-shock response system maybe
devised to achieve
‣ a very small number of unfolded proteins
‣ minimal complexity (no feedback necessary)
❖ E.g. over-expressing chaperones & eliminating feedback
141. Is the Wild Type Optimal?
❖ Observation: A hypothetical heat-shock response system maybe
devised to achieve
‣ a very small number of unfolded proteins
‣ minimal complexity (no feedback necessary)
❖ E.g. over-expressing chaperones & eliminating feedback
❖ However… chaperone over-expression
‣ a high metabolic cost
‣ is toxic to the cell
142. Is the Wild Type Optimal?
❖ Observation: A hypothetical heat-shock response system maybe
devised to achieve
‣ a very small number of unfolded proteins
‣ minimal complexity (no feedback necessary)
❖ E.g. over-expressing chaperones & eliminating feedback
❖ However… chaperone over-expression
‣ a high metabolic cost
‣ is toxic to the cell
❖ The existing design appears to achieve a tradeoff: good folding
achieved with minimal number of chaperones
143. Is the Wild Type Optimal?
❖ Observation: A hypothetical heat-shock response system maybe
devised to achieve
‣ a very small number of unfolded proteins
‣ minimal complexity (no feedback necessary)
❖ E.g. over-expressing chaperones & eliminating feedback
❖ However… chaperone over-expression
‣ a high metabolic cost
‣ is toxic to the cell
❖ The existing design appears to achieve a tradeoff: good folding
achieved with minimal number of chaperones
❖ How optimal is the WT design?
144.
145. ❖ A well-designed system would be configured to
‣ minimize unfolded proteins
‣ minimize the chaperones used
146. ❖ A well-designed system would be configured to
‣ minimize unfolded proteins
‣ minimize the chaperones used
❖ A performance index that captures how well this is achieved:
α reflects the relative importance between unfolded proteins and DnaK
147. ❖ A well-designed system would be configured to
‣ minimize unfolded proteins
‣ minimize the chaperones used
❖ A performance index that captures how well this is achieved:
α reflects the relative importance between unfolded proteins and DnaK
❖ depends on the system parameters . An optimally designed
θ
system would minimize
€
148. ❖ A well-designed system would be configured to
‣ minimize unfolded proteins
‣ minimize the chaperones used
❖ A performance index that captures how well this is achieved:
α reflects the relative importance between unfolded proteins and DnaK
❖ depends on the system parameters . An optimally designed
θ
system would minimize
❖ We solve this optimization problem in silico:
€
149.
150. • For a fixed α, the optimal solution yields a single optimal point:
t1
Unfolded proteins : ∫ [P un ]2 dt
t0
€
t1
2
cost of chaperones : ∫ [DnaK ] dt
t0
€
151. • For a fixed α, the optimal solution yields a single optimal point:
t1
Unfolded proteins : ∫ [P un ]2 dt
t0
€
t1
2
cost of chaperones : ∫ [DnaK ] dt
t0
• If we solve the optimization for all α>0, we get an optimal curv
€
t1
Unfolded proteins : ∫ [P un ]2 dt
t0
Non-optimal
Optimal designs
€
unachievable
t1
2
cost of chaperones : ∫ [DnaK ] dt
t0
152. Pareto Optimal Design of the
Heat Shock System
100
P areto O ptimal curve
80
Cost of u nfolded proteins
60
40
20
W ild type heat shock
0
10 11 12
t1
Cost of chape rones ∫ [DnaK ]2 dt
t0
153. Pareto Optimal Design of the
Heat Shock System
100
P areto O ptimal curve
80
Cost of u nfolded proteins
various nonoptimal values
60 of parameters
40
20
W ild type heat shock
0
10 11 12
t1
Cost of chape rones ∫ [DnaK ]2 dt
t0
154.
155. 1
Sensitivity of DnaK to model parameters
Sensitivity of DnaK 10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
100 200 300 400 500 600 700 800
Time (min)
156. 1
Sensitivity of DnaK to model parameters
10
The complex architecture is a necessary
Sensitivity of DnaK
0
10
outcome of robustness and performance
-1
10
requirements to survive heat-shock
-2
10
-3
10
-4
10
-5
10
100 200 300 400 500 600 700 800
Time (min)
158. Bacterial Chemotaxis
E. coli must swim towards
nutrients or away from repellants
Bacteria are too small to sense
spatial gradients
Instead they rely on a very
effective stochastic strategy Movie by
P. Cluzel
Run and tumble:
Run
Swim with a constant direction (runs)
Changing their direction at random times
(tumbles)
Frequency of tumbling depends on the
Tumble
sensed concentration
Correlation between swimming behavior and flagellar
rotation in E. coli (Cell Project)
159. Bacterial Chemotaxis
E. coli must swim towards
nutrients or away from repellants
Bacteria are too small to sense
spatial gradients
Instead they rely on a very
effective stochastic strategy Movie by
P. Cluzel
Run and tumble:
Run
Swim with a constant direction (runs)
Changing their direction at random times
(tumbles)
Frequency of tumbling depends on the
Tumble
sensed concentration
Correlation between swimming behavior and flagellar
rotation in E. coli (Cell Project)
161. Optimotaxis
Agents mimics bacteria chemotactic Advantages
behavior with the goal of:
Agents simplicity, low cost
Finding the maximum of a measured
quantity; or Increasedprobability of finding the
Finding the spatial distribution of a global maximum due to randomness
measured quantity. Robustness to exogenous
disturbances in the agents orientation.
Chemical plume from BP-Amoco refinery explosion
[courtesy of Los Alamos National Laboratory]
Flapping wing
Agents features Micro aerial vehicle (MAV)
[courtesy of K. Jones, NPS]
Constant velocity
No position or velocity sensors required
No communication needed
Agents can be “seen” by a supervisor
162. Simulation Results Exponential turning rate model
Different stages in optimotaxis in the presence of two maxima
Mesquita et. al., Hybrid Systems: Computation and Control, No. 4981 in Lect. Notes in Comput. Science, 2008.
163. Conclusions
❖ Feedback regulation mechanisms are ubiquitous
❖ A dynamical-systems and control approach can
‣ Bring out the dynamic nature of biochemical interactions
‣ Explain interactions in the context of regulation
‣ Identify functional biological modules
❖ Control theoretic notions
‣ Reveal structural constraints on the dynamics
‣ Structural constraints impose functional requirements on
biological modules
164. ❖ A systems approach enhances our understanding of biological
complexity
‣ Notions such as robustness, adaptation, amplification, isolation, and
nonlinearity are required for a deeper understanding of biological
function
❖ Leads to a better understanding of the trajectory of disease
‣ suggest more effective courses of treatment
❖ Many similarities with engineering systems
❖ New challenges and opportunities for dynamics and control scientists
165. Acknowledgement
❖ Calcium homeostasis: Hana El-Samad (UCSF), Jess Goff (NADC)
❖ Heat Shock: Hana El-Samad (UCSF), Carol Gross (UCSF), John
Doyle (Caltech), Hiro Kurata (KIT, Japan)
❖ UAV search (Joao Hespanha, Alexandre Mesquita (UCSB))
❖ Funding:
‣ National Science Foundation
Notas del editor
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in spite of environmental variations and disturbances\n Claude Bernard (1865)\n Recognized the prevalence of regulatory processes within the organisms (fixite du milieu interieur)\n Walter Cannon (1929)\n Coined the term “homeostasis” to describe the way by which the physical and chemical properties of a living organism are controlled—Wisdom of the Body\n Norbert Wiener (1948)\n Launched an important attempt for interdisciplinary coordination between system the oryand the biological sciences—Cybernetics.\n
in spite of environmental variations and disturbances\n Claude Bernard (1865)\n Recognized the prevalence of regulatory processes within the organisms (fixite du milieu interieur)\n Walter Cannon (1929)\n Coined the term “homeostasis” to describe the way by which the physical and chemical properties of a living organism are controlled—Wisdom of the Body\n Norbert Wiener (1948)\n Launched an important attempt for interdisciplinary coordination between system the oryand the biological sciences—Cybernetics.\n
in spite of environmental variations and disturbances\n Claude Bernard (1865)\n Recognized the prevalence of regulatory processes within the organisms (fixite du milieu interieur)\n Walter Cannon (1929)\n Coined the term “homeostasis” to describe the way by which the physical and chemical properties of a living organism are controlled—Wisdom of the Body\n Norbert Wiener (1948)\n Launched an important attempt for interdisciplinary coordination between system the oryand the biological sciences—Cybernetics.\n
in spite of environmental variations and disturbances\n Claude Bernard (1865)\n Recognized the prevalence of regulatory processes within the organisms (fixite du milieu interieur)\n Walter Cannon (1929)\n Coined the term “homeostasis” to describe the way by which the physical and chemical properties of a living organism are controlled—Wisdom of the Body\n Norbert Wiener (1948)\n Launched an important attempt for interdisciplinary coordination between system the oryand the biological sciences—Cybernetics.\n
99% of all calcium is in the skeleton (1.5% of body weight)\n1% in body fluids. Blood coagulation, muscle contraction, nerve function\n
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(daily need is typically less than 20g/day)\n (up to 50 additional g/day)\n
(daily need is typically less than 20g/day)\n (up to 50 additional g/day)\n
(daily need is typically less than 20g/day)\n (up to 50 additional g/day)\n
(daily need is typically less than 20g/day)\n (up to 50 additional g/day)\n
(daily need is typically less than 20g/day)\n (up to 50 additional g/day)\n
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RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n
RNAP slides rapidly along DNA…Latches tightly when it encounters the promoter (sequence of nucleotides signifying start region)…Subunit of RNAP called sigma factor recognises promoter sequence…Opens up double helix, exposes nucleotides…One of the two DNA strands acts as a template for base pairing\nBy incoming ribonucleatides (AGCU). A medium sized gene ~1500 nucleotide pairs requires 50 seconds for transcription.\nThere maybe 15 RNAP for on one gene. Error rate is 10^-4 compared to 10^-7 in DNA replication.\nRNAP acts as a reading head.\n