Calcium

Invented by Nature, Rediscovered by Man:
Feedback Control Systems in Biology and Engineering

Mustafa Khammash
University of California, Santa Barbara

Outline
❖ Feedback control at the system level
‣ Calcium homeostasis in mammals

Outline

❖ Feedback control at the molecular level
‣ The bacterial Heat-Shock Response

Outline


❖ A Bio-Inspired Engineering Application
‣ Search strategies for unmanned aerial vehicles

Outline



❖ Challenges and opportunities

Outline



❖ Challenges and opportunities
❖ Conclusions

Feedback Control at the System Level
Calcium homeostasis in mammals

Homeostasis
❖ "Homeostasis" is derived from the Greek words for "same" and
"steady."

4

Homeostasis
"steady."

❖ Refers to ways the body acts to maintain a stable internal
environment.

4

Homeostasis
"steady."

environment.

❖ The body is endowed with a multitude of automatic mechanisms of
feedback that counteract influences tending toward disequilibrium.

4

Homeostasis
"steady."

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

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

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

❖ 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

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

Ca Clearance Rate
100

90


70

60


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)

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

Ca Clearance Rate
100

90


70

60


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)

❖ 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

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

Calcium Flow

Milk, fetus

Formation Filtration
Bone Calcium pool Kidney
Resorption reabsorption

Secretion Absorption

Intestine

Mathematical Modeling of [Ca]

Plasma


Ca Total Supply Rate
VT (g/day)

Intestinal Absorption
Plasma
Bone Resorption


Ca Total Supply Rate Total Ca Clearance Rate
VT (g/day) Vcl (g/day)

Plasma Milk, fetus, urine, etc.
Bone Resorption


Ca Total Supply Rate Total Ca Clearance Rate
VT (g/day) Vcl (g/day)

Plasma Milk, fetus, urine, etc.
Bone Resorption

Vol = Plasma Volume (l)
[Ca]p = Plasma Concentration (g/l)

in block diagram form...

1 t
[Ca]p = (VT − Vcl )dτ
Vol 0

Vcl
-
VT + k

Vcl
Set point e VT -
+
+ Control
-

e = error (g/l)

what is f (·) ?

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.

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

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

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:

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.

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

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?

❖ 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:

Hormonal Regulation

The Parathyroid Gland monitors blood
calcium and secretes Parathyroid Hormone
(PTH) in proportion to [Ca] deficiency

Hormonal Regulation


PTH stimulates renal calcium
reabsorption and bone resorption

Hormonal Regulation



(1,25 OH2 D3) Hormone stimulates
calcium absorption from the intestine

Hormonal Regulation



(1,25 OH2 D3) Hormone stimulates
calcium absorption from the intestine

Bone resporption and intestinal absorption
account for the entire calcium supply

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]:

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

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

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

Control at the Molecular Level
Bacterial Heat Shock Response

Gene Expression

Transcription
mRNA

start
DNA

end
promoter
RNA polymerase

Gene Expression

Transcription
mRNA

start
DNA
mRNA
end
promoter
RNA polymerase DNA

Gene Expression

Translation proteins

mRNA
Transcription ribosomes
mRNA

start
DNA
mRNA
end
promoter
RNA polymerase DNA

Gene Expression


mRNA
mRNA
protein
start
DNA
mRNA
end
promoter
RNA polymerase DNA

Gene Expression


Central Dogma of
Molecular Biology

mRNA
mRNA
protein
start
DNA
mRNA
end
promoter
RNA polymerase DNA

Cell

Temp
cell

Temp
environ

Cell

Unfolded
Proteins

Temp
cell
Folded
Proteins
Temp
environ

Cell

Unfolded
Proteins
Aggregates

Temp
cell
Folded
Proteins
Temp
environ

Cell
Loss of Protein
Function

Unfolded
Proteins
Aggregates

Temp
cell
Folded
Proteins
Temp
environ

Cell
Loss of Protein Network
Function failure

Unfolded
Proteins
Aggregates

Temp
cell
Folded
Proteins
Temp
environ

Cell
Loss of Protein Network
Function failure

Death

Unfolded
Proteins
Aggregates

Temp
cell
Folded
Proteins
Temp
environ

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.


❖ 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

Function of the Heat-Shock Proteins

I. Protein Folding
DnaK/J GroEL/
GroES

Unfolded/partially folded
Proteins

Folded
Proteins

I. Protein Folding
DnaK/J GroEL/
GroES

Unfolded/partially folded
Proteins

II. Protein Degradation

Proteases

Folded
Amino Acids Proteins
Proteins Aggregates

Heat-Shock Gene Transcription

start DNA end
hsp1 hsp2

promoter terminator

Heat-Shock Gene Transcription
factor

start DNA end
hsp1 hsp2

promoter terminator

RNA Polymerase

mRNA Translation

Heat-Shock Proteins

mRNA

ribosomes

Regulation of the Heat Shock Response

Tight regulation of σ32 at 3 levels

Synthesis Activity Stability

Feedforward Feedback

I. Regulation of σ32 Synthesis

mRNA

mRNA

At low temperature, mRNA has a secondary structure

mRNA

mRNA


mRNA

Heat

mRNA


mRNA

Translation
Heat

mRNA

II. Regulation of σ32 Activity: A Feedback Mechanism


RNAP hsp1 hsp2


RNAP hsp1 hsp2

Transcription & Translation

Chaperones


RNAP hsp1 hsp2


Chaperones
Heat

III. Regulation of σ32 Degradation
FtsH degrades sigma-32 only when bound to chaperones

RNAP hsp1 hsp2


Chaperones
Heat


RNAP hsp1 hsp2


Proteases Chaperones
Heat
FtsH

FtsH
FtsH


RNAP hsp1 hsp2


Proteases Chaperones
Heat
FtsH

FtsH
FtsH
FtsH

Disturbance

FF sensor

FB
Control sensor

Plant

Actuator
A control theorist’s view.
What is the relation to the HS system?

σ mRNA Heat

FF sensor

sensor
Control
Plant
RNAP

FtsH
Actuator
RNAP
hsp1 hsp2

Mathematical Model

Protein
Synthesis

Binding
Equations

Mass -
Balance
Equations

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)

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



Analysis Tools
‣ Dynamic analysis and simulations



Analysis Tools
‣ Sensitivity/robustness analysis



Analysis Tools
‣ Sum-of-Squares tools



Analysis Tools
‣ Sum-of-Squares tools
‣ Optimal control

600

450
Total σ32

300

150 Wild type

0
0 10 20
0
Time (min)

30o 42o

600

450
Total σ32

300

150 Wild type
No feedforward
0
0 10 20
0
Time (min)

30o 42o

600

450
Total σ32

300

150 Wild type
No feedforward
0
0 10 20
0 No DnaK interaction
Time (min)

30o 42o

600

450
Total σ32

300

150 Wild type
No feedforward
0
0 10 20
Time (min)
Constitutive σ32 degradation
30o 42o

600

450
Total σ32

300

150 Wild type
No feedforward
0
0 10 20
Time (min)
30o 42o Low σ32 flux

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)
30 42o Low σ32 flux
o

600 2400
DnaK
450
Total σ32 1600

300 1600

1200
150 Wild type
No feedforward
800
0
0 10 20
20
0
Time (min)
o
8

6

4
Free σ32

2

0 10 20
0
0
Time (min)

600 2400
DnaK
450
Total σ32 1600

300 1600

1200
150 Wild type
No feedforward
800
0
0 10 20
20
0
Time (min)
o
8 E+ 06

6

4
Free σ32 Unfolded Proteins

2

0 0
0 10 20
0 0 10 20
Time (min) Time (min)


– Disturbance feedforward


– Disturbance feedforward Fast response



– Activity feedback loop



– Activity feedback loop Robustness, efficiency




– Degradation feedback loop




– Degradation feedback loop Fast response, noise suppression





– High sigma-32 flux





– High sigma-32 flux Fast response

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)

devised to achieve

❖ E.g. over-expressing chaperones & eliminating feedback

devised to achieve

❖ However… chaperone over-expression
‣ a high metabolic cost
‣ is toxic to the cell

devised to achieve


❖ The existing design appears to achieve a tradeoff: good folding
achieved with minimal number of chaperones

devised to achieve


❖ The existing design appears to achieve a tradeoff: good folding
achieved with minimal number of chaperones

❖ How optimal is the WT design?

❖ 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

€




❖ depends on the system parameters . An optimally designed
θ
system would minimize

❖ We solve this optimization problem in silico:
€

• 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

€

• For a fixed α, the optimal solution yields a single optimal point:

t1

t0

€
t1
2
t0

• If we solve the optimization for all α>0, we get an optimal curv
€

t1
t0

Non-optimal
Optimal designs
€
unachievable
t1
2
t0

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

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

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)

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)

A Bio-Inspired Engineering Application
Search strategies for unmanned aerial vehicles

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)

The Flagellar Motor

Keiichi NAMBA Francis et al., 1994

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

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.

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

❖ 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

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

Calcium

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