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Exact results for optimal phenotypic switching rates. 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. - PowerPoint PPT Presentation
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Exact results for optimal phenotypic switching
rates 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
Reluga 2005Salathé et al. 2009
Kussell and Leibler, 2005
Ishii et al. 1989
Thattai and van Oudenaarden, 2004
Lachmann and Jablonka, 1996
...optimal rate of switching is approximately equal to the rate of environmental change.
...environment that periodically fluctuates between two states
...what about when environments are different?
Problem setup...
Periodic environments, “A” and “B”. We can think of one of the environments as a period of antimicrobial treatment
AEA AE
BBE
A BEB
Problem setup, in maths
Long term max growthrate:
where
kk
kkM
AB
AA
EA
kk
kkM
BB
BA
EB
BAAE
AA n kn kn
dt
dn
Ank nk ndt
dnBB
EB
B x
x EMdt
d
x(0)e...eeeeex(t)
timesq
TMTMTMTMTMTM BBEAAEBBEAAEBBEAAE
BA
TMTM
TT
)e(elnr
BBEAAE
The Solution...
where
))(4k(4k),,T,T(k,
),,T,T(k,4k4k
e
2B
22A
2BABA
2
BABA2B
22A
2
)T(T2
1k))(T(T BBAABA
(k,TA,TB,A,B) (4k 2 AB)sAsB 4k2 A2 4k2 B
2cAcB
and
si sinh(Ti
24k2 i
2 ),c i cosh(Ti
24k2 i
2 )
Gaal, Pitchford, Wood. Genetics Vol. 184, 1113
Some definitions
Universal switching rateBA E
BE
A k
Fitness of fitter types is the same
BEA
AEB
Differing fitness penalties in the two different environments
AT Time spent in environment ATime spent in environment B
BTGaal, Pitchford, Wood. Genetics Vol. 184, 1113
Key result – not switching is good
10T50T 20T
Gaal, Pitchford, Wood. Genetics Vol. 184, 1113
Key result – discontinuous change
r
k
0
Max
Max
5.0
Max
0.1
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 GaalJon Pitchford Marjan van der Woude
