diffusion, and how it is used or how stupid organisms do...

University of UtahMathematical Biology

theImagine Possibilities

Diffusion, and How it is Usedor

How Stupid Organisms Do MathJ. P. Keener

Department of MathematicsUniversity of Utah

Diffusion, and How it is UsedorHow Stupid Organisms Do Math – p.1/19



Introduction

Fick’s law: Small molecules undergo a random walk. When thereare a large number of these molecules, their motion can bedescribed by

J = − D∂C

∂x

Conservation:∂C

∂t+

∂J

∂x= 0

leading to the Diffusion Equation

∂C

∂t=

∂

∂x(D

∂C

∂x).




Introduction


J = − D∂C

∂x

molecular flux,

Conservation:∂C

∂t+

∂J

∂x= 0


∂C

∂t=

∂

∂x(D

∂C

∂x).




Introduction


J = − D∂C

∂x

molecular flux, diffusion coefficient,

Conservation:∂C

∂t+

∂J

∂x= 0


∂C

∂t=

∂

∂x(D

∂C

∂x).




Introduction


J = − D∂C

∂x

molecular flux, diffusion coefficient, concentration gradient .

Conservation:∂C

∂t+

∂J

∂x= 0


∂C

∂t=

∂

∂x(D

∂C

∂x).




Introduction


J = − D∂C

∂x

molecular flux, diffusion coefficient, concentration gradient.Conservation:

∂C

∂t+

∂J

∂x= 0


∂C

∂t=

∂

∂x(D

∂C

∂x).




Basic Consequences - I

Diffusion in a tube fed by a reservoir

C(x, t) = f(x2

Dt)

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

C




Basic Consequences - II

Diffusion time:t = x2

Dfor hydrogen (D = 10−5cm2/s).

x t Example10 nm 100 ns cell membrane1 µm 1 ms mitochondrion10 µm 100 ms mammalian cell

100 µm 10 s diameter of muscle fiber250 µm 60 s radius of squid giant axon1 mm 16.7 min half-thickness of frog sartorius muscle2 mm 1.1h half-thickness of lens in the eye5 mm 6.9 h radius of mature ovarian follicle2 cm 2.6 d thickness of ventricular myocardium1 m 31.7 yrs length of sciatic nerve




Basic Consequences - II

Diffusion across a membrane

� ��

V

C2

1 V2L

C1

J =AD

L(C1 − C2)

then

d

dt(V1C1) =

AD

L(C2 − C1), V1C1 + V2C2 = (V1 + V2)C0




Carrier Mediated Diffusion

Problem: If glucose only moves down its gradient, there mustalways be more glucose in the blood than in cells, or else cellswill lose their glucose.

� � � � � � � � � � � � � � � � � � � � ��






� � � � � � � � � � � � � � � � � � � � ��

For this system,

J = Jmax

ge − gi

(ge + K)( gi

K+ 1)






P

� � � � � � � � � � � � � � � � � � � � ��

Solution: Immediately phosphorylate internal glucose, settinggi = 0 so that

J = Jmax

ge

ge + K



theImagine Possibilities Oxygen and Carbon Dioxide

Transport

Problem: If oxygen and carbon dioxide move into and out of theblood by diffusion, their concentrations cannot be very high (andno large organisms could exist.)

CO

COO CO2O222

O2 2

In Tissue In Lungs

CO2 O2

Chemical reactions that help enormously:

CO2 + H2O→

← HCO+3

+ H− Hb + 4O2

→

← Hb(O2)4

Hydrogen competes with oxygen for hemoglobin binding.




Transport


−HCO3

−HCO3

CO2 CO2

CO2

In Tissue In Lungs

O2

O2

O2 −

−

CO2 O2

H

2

H

HbO 42HbO 4


CO2 + H2O→

← HCO+3

+ H− Hb + 4O2

→

← Hb(O2)4

Hydrogen competes with oxygen for hemoglobin binding.




Transport


CO2

HCO 3

In Tissue In Lungs

O2

4HbH

O2

O 2

CO 2

CO2CO2 O

2

2

HbO 4−

HCO3


CO2 + H2O→

← HCO+3

+ H− Hb + 4O2

→

← Hb(O2)4

Hydrogen competes with oxygen for hemoglobin binding.Diffusion, and How it is UsedorHow Stupid Organisms Do Math – p.7/19



Quorum Sensing

Quorum sensing: The ability of bacteria to respond to theirpopulation size. Question: How do bacteria conduct a census?

Solution: Autoinducer (HSL)- a freely diffusing chemical withauto-catalytic (positive feedback) production.

3−oxo−C12−HSL

LasR

rsaL

LasI

RsaL

LasR

C4−HSL

LasR

Vfr

rhlR

RhlR

RhlR

RhlI

rhlI

lasR

lasI




Quorum Sensing

Quorum sensing: The ability of bacteria to respond to theirpopulation size. Question: How do bacteria conduct a census?

E

A

E

A

Solution: Autoinducer (HSL)- a freely diffusing chemical withauto-catalytic (positive feedback) production.

3−oxo−C12−HSL

LasR

rsaL

LasI

RsaL

LasR

C4−HSL

LasR

Vfr

rhlR

RhlR

RhlR

RhlI

rhlI

lasR

lasI




Quorum Sensing-II

E

A

E

A

dA

dt= F (A) + δ(E −A)

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

F(A

)




Quorum Sensing-II

E

A

E

A

dA

dt= F (A) + δ(E −A)

rate of change of A,

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

F(A

)




Quorum Sensing-II

E

A

E

A

dA

dt= F (A) + δ(E −A)

production of A, production rate,

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

F(A

)




Quorum Sensing-II

E

A

E

A

dA

dt= F (A) + δ(E −A)

production of A, production rate, diffusive loss .

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

F(A

)




Quorum Sensing-III

Extracellular Autoinducer E:

E

A

dE

dt= − kEE + δ(A− E)

.




Quorum Sensing-III


E

A

dE


rate of change of E, .




Quorum Sensing-III


E

A

dE


rate of change of E, degradation rate, .




Quorum Sensing-III


E

A

dE


rate of change of E, degradation rate, diffusive source, .




Quorum Sensing-III


E

A

(1− ρ) (dE

dt+ KEE) = ρ δ(A− E)

rate of change of E, degradation rate, diffusive source,density dependence .




Quorum Sensing-IV

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2

3

4

5

Internal Autoinducer

Ext

erna

l aut

oind

ucer

A nullclineE nullcline




Length Detection




Observations

• Flagella grow at a velocity that decreases as they get longer.

• If a flagellum is cut off, it will regrow at the same velocity aswhen it first grew.

Question: How does the bacterium "know" how long its flagellaare?




Observations

• Flagella grow at a velocity that decreases as they get longer.• If a flagellum is cut off, it will regrow at the same velocity as

when it first grew.

Question: How does the bacterium "know" how long its flagellaare?




How Do Flagella Grow?

Stage 1: Secretion of Flagellin (FliC)

Step 1

FliI

FliJ

� � ��

� ��

Cring

MS ring

CM FlhA FlhB

C

N

FliH

membranecomponents






� � ��

� ��

FliJ

Step 2

N

C

Cring

MS ring

CM FlhA FlhB

membranecomponents

FliH

FliI






� � ��

� ��

N

C

Step 3

Cring

MS ring

CM FlhA FlhB

FliJ

ATP ADP+Pi

FliH

membranecomponents

FliI






� � ��

� ��

C

Step 4

FliJ

N

Cring

MS ring

CM FlhA FlhB

membranecomponents

FliI

FliH





Stage 2: Diffusion and polymerization

! ! !" " "# # #$ $ $ $ $ $ $$ $ $ $ $ $ $% % % % % % %% % % % % % %

A good approximation

J ≈1

KJ + LD

≈

D

Lfor large L(length dependence!)




Length Detection

The protein FlgM has three important properties:• It is secreted during filament growth;• It inhibits its own production (negative feedback);• It inhibits the production of Flagellin.




Tracking Concentrations

FlgM (M ):

dM

dt= rate of production− rate of secretion

Flagellin (F ):

dF

dt= rate of production− rate of secretion

Filament Length (L):

dL

dt= β

F

M + FJ

withJ =

1

KJ + LD




Tracking Concentrations

FlgM (M ):

dM

dt=

K∗

KM + M− α

F

F + MJ

Flagellin (F ):

dF

dt=

K∗

KM + M− α

M

F + MJ

Filament Length (L):

dL

dt= β

F

M + FJ

withJ =

1

KJ + LDDiffusion, and How it is UsedorHow Stupid Organisms Do Math – p.17/19



Filament Growth

0 200 400 600 800 1000 1200 14000

5

10

15

20

25Filament Length vs Time

Leng

th (m

icro

ns)

Time (minutes)0 200 400 600 800 1000 1200 1400

0

500

1000

1500

2000

2500

3000

3500

4000

Time (minutes)

Intracellular FlgM and FliC

Num

ber o

f Mol

ecul

es

• FlgM concentration is initially large. When secretion begins,FlgM concentration drops, producing FliC and more FlgM.

• As filament length grows, secretion slows, FlgMconcentration increases, shutting off FliC and FlgMproduction.

• If filament is suddenly shortened, secretion suddenlyincreases, reinitiating the grow phase.




Filament Growth

0 200 400 600 800 1000 1200 14000

5

10

15

20

25Filament Length vs Time

Leng

th (m

icro

ns)

Time (minutes)0 200 400 600 800 1000 1200 1400

0

500

1000

1500

2000

2500

3000

3500

4000

Time (minutes)

Intracellular FlgM and FliC

Num

ber o

f Mol

ecul

es

• FlgM concentration is initially large. When secretion begins,FlgM concentration drops, producing FliC and more FlgM.

• As filament length grows, secretion slows, FlgMconcentration increases, shutting off FliC and FlgMproduction.

• If filament is suddenly shortened, secretion suddenlyincreases, reinitiating the grow phase.Diffusion, and How it is UsedorHow Stupid Organisms Do Math – p.18/19



Acknowledgments

Collaborators• Jack Dockery, Montana State University (quorum sensing)

Notes• Funding for research provided by a grant from the NSF.• No Microsoft products were used or harmed during the

production of this talk.

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