# Exact results for optimal phenotypic switching rates A Jamie Wood,

### Exact results for optimal phenotypic switching rates

A Jamie Wood,

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: