Information limits of the accuracy of directed motion of biological cells

1. Igor Šegota
Department of Physics
Cornell University, USA
Information limits of the accuracy
of directed motion of biological cells

2. Two approaches to biological models
(b) Experimental data-driven statistics
Modeling accuracy limits of various
biological processes
Edward M. Purcell Howard Berg
(a) Dynamical systems / differential equations
Detailed descriptions but need to estimate
many unknown parameters
Gutenkunst et al. PLoS Comp. Biol. (2007)
Focus of this work
Berg, Purcell Biophys J. (1977)

3. How do cells measure concentration
of extracellular molecules?
cells (~ 1-10 µm)
receptors, molecules (~ 0.1-1 nm)
Nrec ~105 receptors / cells
unbound
receptor
R*
cell
molecule
concentration c
molecule bound
to a receptor R
kon
Cells have receptors that bind molecules (ligands):
koff

4. change in molecule
number per time
= – molecule-receptor
binding rate
molecule-receptor
dissociation rate
dt
dc
R
koff
*
oncR
k
= –
kon
koff
R
R*
c
Stationary state
rec
*
N
R
R 

on
off
D k
/
k
K 
0
dt
dc

D
rec
K
c
c
N
R


D
K
c
c
p


Number of bound receptors:
Probability of receptor
being bound to a molecule:
How do cells measure concentration
of extracellular molecules?

5. Limits of concentration measurements due to
molecular diffusion
concentration c
random walk,
diffusion const. D
average number of
molecules N = cr3
correlation time
τC ~ r2/D
Δc
c
=
1
nmeas
ΔN
N
=
τc
τmeas
1
N
~
1
Drcτmeas
Measurement
uncertainty during
time τmeas:
H Berg, EM Purcell, Biophys. J. (1977)
r
~ 1 nM
~ 400 µm2/s
~ 1 µm
~ 5%
~ 1 s
“Perfect instrument”

6. How do cells measure concentration gradients of
extracellular molecules?
Eukaryotic cells (~ 10 µm)
Measures concentration across cell body
position
Number of bound receptors at position x
-r 0 r
time
Total number of bound receptors
Bacteria (~ 1 µm)
Measures concentration change over time
c1, t1
c2, t2
c3, t3
c4, t4
c5, t5
Ñc®
¶c
¶t

7. Concentration gradient is encoded in the spatial
distribution of bound receptors
Eukaryotic cells
- How can they measure gradients?
R
2
R
1
Each receptor modelled as a
Bernoulli event: or
D
2
,
1
2
,
1
rec
2
,
1
K
c
c
2
N
R


 2
D
2
,
1
D
2
,
1
rec
2
R
K
c
K
c
2
N
2
,
1



p( )
p( )p( )
average number of
bound receptors
variance of the
number of bound
receptors
For each cell half: binomial distribution

8. Cells detect concentration gradients with as few as
4 receptors difference between front and back !
Eukaryotic cells
- How can they measure gradients?
R
2
R
1
Shot noise regime:
ΔR ± σΔR = 4 ± 15
Detection limit, experiment with
Dictyostelium discoideum:
c = 0.5 nM,
c = 3.3×10-3 nM/µm,
KD = 100 nM, r = 5 µm, Nrec = 5×104
L. Song et al., Eur. J. Cell. Biol. (2006)
 2
D
D
rec
2
R
K
c
cK
N



Difference in number of bound
receptors in the limit of shallow
gradients:
 2
D
D
rec
K
c
c
r
K
4
N
R





9. Concentration gradient measurement accuracy can be
quantified using a Signal-to-Noise ratio
Signal-to-Noise Ratio:
R
R
SNR




0
0.1
0.2
0.3
0.4
0.5
0.001 0.01 0.1 1 10 100 1000
average concentration, c / KD
rec
N
4
SNR

c
c
r


Special case: % gradient across cell body = const. =
 
D
D
rec
K
c
c
c
r
K
4
N




10. Concentration gradient measurement accuracy can be
quantified using Mutual information
Entropy as information measure: 


x
2 )
x
(
p
log
)
x
(
p
)
X
(
H Shannon C, Bell Sys.
Tech. J. (1967)
p(x)
H = max H = 0
Relative entropy
 


y x
2 )
y
|
x
(
p
log
)
y
|
x
(
p
)
y
(
p
)
Y
|
X
(
H
= degree of “sharpness” of
probability distribution of X, when
we knowY (on average)
Mutual information:
0
)
Y
|
X
(
H
)
X
(
H
)
Y
,
X
(
I 


= difference in “sharpness” of
two probability distributions
= “amount” of information of X
contained inY

11. Mutual information can be used to define a physical
limit to gradient detection
conditional distribution of
bound receptors
p(Y | θg) = p( , , ,…, | θg)
distribution of
gradient direction
p(θg)
-π π
conditional distribution
of cell response
p(θres | θg)
π
-π
 
g
res
tot ,
I 

Total information: can be
measured
 
2
D
D
rec
g
ext
K
c
c
r
c
K
2
ln
4
N
,
Y
I 











Fuller D et al. PNAS (2011)
External information: can be
calculated
Markov chain: θg →Y → θres tot
ext I
I  data processing
inequality

12. Measuring information in cell chemotaxis
Our experimental model:
Eukaryotic single-cellular organism: Dictyostelium discoideum amoeba

13. Measuring total information in cell chemotaxis
• detect gradients of folid acid and cyclic adenosine-monophosphate (cAMP)
Bacteria → folic acid → Dicty
Jeremy Pickett-Heaps,
Uni of Melbourne (2013)
10 µm
Dictyostelium → cAMP → Dictyostelium
Duane Loh & Albert Bae
Cornell University (2009)
well studied, as opposed to folic acid
chemotaxis

14. Measuring total information in cell chemotaxis
gradient established by diffusion through gel
design Prof. M.Wu (Cornell Bioeng.)
100 µm
gradient
res

N = 800
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
-180
-150
-120
-90
-60
-30
0
30
60
90
120
150
Relative
fraction
or
p(θ
res
|θ
grad
)
θres (°)
Folic acid concentration gradient:

15. Iext
Itot
Total information exceeded its “theoretical maximum”
(external information) !
0.6%
0.00
0.05
0.10
0.15
0.20
0.1 1 10 100 1000 10000100000
mutual
information
[bits]
average concentration [nM]
Segota et al., J. Roy. Soc. Interface (2013)
tot
ext I
I 
0.00
0.05
0.10
0.15
100 1000 10000 100000
mutual
information
[bits]
average concentration [nM]
0.32 nM/µm 1.61 nM/µm 3.21 nM/µm
Itot
Iext
KD = 150 nM Nrec = 60 000
Wurster et al., J. Bacteriol. (1981)

16. Total information exceeded its “theoretical maximum”
(external information) !
0.00
0.05
0.10
0.15
0.20
0.1 1 10 100 1000 10000100000
mutual
information
[bits]
average concentration [nM]
Itot
Iext
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.1 1 10 100 1000 10000100000
mutual
information
[bits]
average concentration [nM]
Itot
Iext
multiple receptor types for folic acid
DeWit J et al., Biochim. Biophys. Acta (1985)
KD [nM] Nrec
450 80,000
70 80,000
17 550
KD [nM] Nrec
50 50
15 1,450
multiple independent measurements
tcorrel
» 5 s Rappel JW, Levine H, PNAS (2011)
tpseudopod
»15-25 s
Nmjerenja
» 3-5

17. Is there a gradient pre-amplifier?
• Second experimental system: Dictyostelium discoideum cells + cAMP gradients
• Extracellular cAMP is degraded by an enzyme phosphodiesterase (PDE)
cAMP
concentration
x

18. Reaction-diffusion model
• Extracellular cAMP is degraded by phosphodiesterase (PDE)
• cAMP and PDE interaction modeled as Michaelis-Menten kinetics:
k1
k-1
k2
cAMP PDE complex PDE
5’AMP
deactivated signal
c p X p
pc
K
k
c
D
t
c
M
2
2
c 




p
D
t
p 2
p



1
2
1
M
k
k
k
K

 
diffusion reaction
equivalent to Debye screening model in electrostatics!
Assuming quasi-stationary state: X
)
k
k
(
pc
k 2
1
1 
 
association complex dissociation

19. Reaction diffusion model and
boundary conditions
• 3D hemisphere (cell) in the middle
model of microfluidic experiment
concentration
is
fixed:
c=c
L
,
p=0
concentration
is
fixed:
c=c
R
,
p=0
0
p
n̂
c
n̂ 







0
p
n̂
c
n̂ 







0
p
p
n̂ 



0
c
n̂ 



Finite element method (FEM)
cAMP concentration [nM]

20. Numerical solution shows there is a parameter range with
preamplification !
estimates of biological
secretion rates
Signal-to-noise ratio (SNR)
R
R
SNR




R
2
R
1
Changing parameters:
1. secretion rate p0
2. cAMP concentration
1%
average
cAMP
concentration
[Kd]
PDE secretion rate
average cAMP concentration [Kd]
PDE secretion rate
signal-to-noise ratio

21. Can we experimentally validate this ?
• We cannot directly measure SNR but can chemotactic index (CI)
gradient
n̂
i
r


i
i
r





















2
SNR
Erf
2
.
0
1
2
SNR
Erf
CI
Empirical relationship

 


i
i
i
i
r
n̂
r
puta
duljina
ukupna
a
gradijent
smjeru
u
pomak
CI 

van Haastert PJM, Postma M, Biophys. J., (2007)
cumulative function of
Normal distribution ~ Erf()
∆R
p(∆R)
0
total path length
gradient displacement

22. Can we experimentally validate this ?
1%
estimates of biological
secretion rates
PDE secretion rate
PDE secretion rate
average
cAMP
concentration
[Kd]
average cAMP concentration [Kd]
chemotactic
index
(CI)

23. Conclusion
• Dictyostelium folic acid chemotaxis is much more accurate than a prediction
from simple receptor-ligand binding model
• It is still an open question how exactly cells acquire so much information
• Mechanism of gradient amplification by secretion of degrading enzymes can
substantially increase the signal-to-noise ratio (SNR)
• Optimal secretion rate of PDE is always > 0 !
• pdsA- mutants (without PDE), respond to smaller range of concentrations (as
predicted by our model)
• Next steps? Experimental tests (CI)