A Jamie Wood,
Bernadett Gaal, Jon Pitchford

Phenotypic switching

Phenotypic switching

Importance
May be very relevant to infection, the cull is the antibiotic, regrowth of nasty phenotype still occurs. NOT a mutation.
When to switch? When not to switch? How has switching evolved? What is optimal switching?

Previous Work
...environment that periodically fluctuates between two states
Ishii et al. 1989
...optimal rate of switching is approximately equal to the rate of environmental change.
Lachmann and Jablonka, 1996
Thattai and van Oudenaarden, 2004
Kussell and Leibler, 2005
...what about when environments are different?
Reluga 2005
Salathé et al. 2009

Problem setup...
Periodic environments, “A” and “B”. We can think of one of the environments as a period of antimicrobial treatment

A
E
A

B
E
A

A
E
B 
B
E
B

Problem setup, in maths dn
A dt dn
B
 
B
E n
B
 k n
B
 k n
A dt
 
A
E n
A
 k n
A
 k n
B
 d dt x

M
E
 x x(t)

     e
M
E
A
T
A e
M
E
B
T
B
   e
M
E
A
T
A
 e
M
E
B
     
T
B ...e
M
E
A
T
A e
M
E
B
T
B x(0)
Long term max growthrate: r
 ln


(e
M
E
A
T
A e
M
E
B
T
B
T
A

T
B
) where
M
E
A




A
A k
 k

B
A k
 k


M
E
B

 
A
B k
 k

B
B k
 k



The Solution...
  
 e
(T
A

T
B
)(
  k)

1
2
(T
A

A

T
B

B
)
4k
2  
A
2
4k
2  
B
2
 
(k, T
A
, T
B
,

A
,

B
)

2
(k, T
A
, T
B
,

A
,

B
)

(4k
2  
A
2
)(4k
2  
B
2
) where
 (k,T
A
,T
B
, 
A
, 
B
)  (4k 2 
A

B
)s
A s
B
 4k 2  
A
2 and s i
 sinh(
T i
2
4k 2  
B
2 c
A c
B
4k 2   i
2 ),c i
 cosh(
T
2 i 4k 2   i
2 )
Gaal, Pitchford, Wood. Genetics Vol. 184, 1113


Some definitions

A
E
A
 k
  
B
E
B
 E
B
A
 E
A
B
   
     
T
A
T
B
Universal switching rate
Fitness of fitter types is the same
Differing fitness penalties in the two different environments
Time spent in environment A
Time spent in environment B
Gaal, Pitchford, Wood. Genetics Vol. 184, 1113

Key result – not switching is good

T

50 T

20 T

10

Gaal, Pitchford, Wood. Genetics Vol. 184, 1113

Key result – discontinuous change
Max r
Max
 
0 .
5
Max
 
0
 
1 .
0 k
Gaal, Pitchford, Wood. Genetics Vol. 184, 1113

More exotic outcomes
Allowing time periods to also vary
Gaal, Pitchford, Wood. Genetics Vol. 184, 1113

A few other ideas
Mathematical framework is well established – can we move to more interesting examples for biology?
Fitness? Is switching directly or indirectly affecting fitness?
Spatial position in a biofilm for instance.
What implications do these results have for evolution of switching?

Thanks to:
Bernadett Gaal
Jon Pitchford
Marjan van der Woude