Stochastic effects for interacting
microbial populations
Rosalind Allen
School of Physics and Astronomy, Edinburgh University
eSI “Stochastic effects in microbial infection”
September 29th 2010
Andrew Free
School of Biological Sciences
Edinburgh University
Eulyn Pagaling
Fiona Strathdee
Bhavin Khatri
Jana Schwarz-Linek
Richard Blythe
Mike Cates
Wilson Poon
Human bodies contain complex microbial communities
Eg intestine contains
~1014 microbes, ~400 species
• Various chemical niches
(fermentation, methanogenesis,
sulphate reduction)
• competition for resources
• interaction with host
• interaction with environment via
immigration and washout
Infecting microbes must compete with
normal flora
R. Ley et al Cell 124, 837–848 (2006)
Germ stories by Kornberg
General questions about microbial communities
• How do complex microbial communities get established?
• How resilient are communities to disturbance (eg antibiotic treatment)
• How likely are invaders to succeed?
• How stochastic are these processes?
Relevant to understanding infection?
Our model system: the Winogradsky column
O2
Aerobic
water
Carbon Cycle
Sulphur Cycle
Organic acids and
CO2 fixed into
organic matter
Sulphur oxidisers
SO42- <- H2S
Anaerobic
water
Cell death
Anaerobic
sediment
Organic acids and
CO2 released by
decomposers
Sulphur reducers
SO4 -> H2S
H2 S
Aim: use this system to learn about microbial community dynamics
Which microbes are present?
Denaturing gradient gel electrophoresis (DGGE)
• Extract DNA from the community
• Use PCR to amplify 16S rRNA gene
fragments ~200bp
• Run on gel, gradient of denaturant
• different sequences stop in different
places
-> fingerprint of the community
“one band = one 16S rRNA gene fragment”
Also analyse community function from
redox gradient top -> bottom
1. How do communities colonise new environments?
Put different communities in the same environment.
Do they develop differently or the same?
Trossachs Lochs
Loch Leven (6 sites)
Blackford pond
Sample after 16 weeks
Blackford Pond
sediment + nutrients
36 sterilised microcosms
Inoculate with different
communities in triplicate
Results: the communities “remember” their origin
Measure similarity between DGGE fingerprints (Bray-Curtis)
-> similarity matrix -> cluster analysis (MDS)
Microcosm communities tend
to cluster according to
geographical origin
But identical communities can give different outcomes
In function (redox)
1
2
3
1
2
3
and community composition
In progress:
Are some aspects of the community more stochastic than others?
Are other aspects more strongly dependent on initial community?
Modelling interacting microbial populations
Example:
Cycling of carbon by methanogens and
methanotrophs:
Methanogens
Carbon dioxide + hydrogen/acetate -> methane
Methanotrophs
Methane + oxygen -> carbon dioxide
A highly simplified model
Waste product of microbe 1 is substrate for microbe 2
Waste product of microbe 2 is substrate for microbe 1
Variables
Microbe population sizes n1 and n2
Substrate concentrations s1 and s2
Parameters
Substrate inflow rates q1, q2
Growth parameters vmax,Km,f for both populations
Death rates m1, m2 for the microbes
Results: “Boom-bust” cycles
(only substrate 1 supplied)
Microbe 1
Microbe 2
• Inflow of substrate 1 causes population boom of microbe 1
• Microbe 1 produces substrate 2
• This causes population boom of microbe 2, accompanied by microbe 1
• Eventually steady state is reached
What happens when we include noise?
Deterministic equations
 
dX
AX
dt
X
Equivalent stochastic equations
is the vector (n1,n2,s1,s2)
 
dX
 A X W
dt

is a Gaussian white noise vector
zero mean, unit variance
W
describes coupling between fluctuations
of substrate and microbial populations
(can derive from Master Equation)
Deterministic
Stochastic
Noise can cause persistent oscillations
To do:
Develop more realistic models for microcosm communities
Can we predict effects of changing environmental conditions?
(eg cellulose)
Conclusions
Microbial community development has significant stochasticity
We’re trying to understand it better using model microcosms
Modelling may help us track down the origin of the variability
How to relate this to infection?
Gut communities may be metabolically simpler than our microcosms
Theoretical models for community dynamics in the gut?
Connection with models of individual species growth and interactions? (eg phase
variation + interspecies interactions…)
Do suitable experimental “microcosm” systems exist?
The End
Growth of a microbial population
Microbe population size n(t)
Substrate concentration s(t)
Waste product concentration w(t)
v s (t )
dn
 n(t ) f / c  max
dt
K m  s (t )
v s (t )
ds
 n(t ) max
dt
K m  s(t )
Vmax = maximal substrate consumption rate / bacterium
Km = substrate concentration for half maximal growth
f = fraction of substrate carbon used for growth
c = carbon / bacterium
dw
ds
 1  f 
dt
dt
Results: “Boom-bust” cycles
(only substrate 1 supplied)
“Boom-bust” dynamics
Microbe 1
Microbe 2
• Inflow of substrate 1 causes population
boom of microbe 1
• Microbe 1 produces substrate 2
• This causes population boom of
microbe 2, accompanied by microbe 1
• Eventually steady state is reached
Substrate 1
Substrate 2
vmax,1 = 24.9 umoles carbon / bug / litre / day
vmax,2 = 5.81 umoles carbon / bug / litre / day
Km,1 = 6.24 umoles carbon / litre
Km,2 = 2.49 umoles carbon / litre
f1 = 0.76
f2 = 0.64
m1 = 0.1 X 109 bugs / litre / day
m2 = 0.1 X 109 bugs / litre / day
q1 = 10 umoles C / litre / day
q2 = 0