Evolution and Regulation of syntrophic symbiosis
Roger Nisbet1, Ross Cunning2, Glenn Ledder3, Erik Muller1, Sabrina Russo3,
Elke Zimmer4
1.
2.
3.
4.
University of California, Santa Barbara, California, USA
Hawaii Institute of Marine Biology, University of Hawaii at Manoa, USA
University of Nebraska, Lincoln, USA
Ibacon GmbH, Rossdorf, Germany
Special thanks to working group on “DEB model for trees” at National Institute for Mathematical and Biological
Synthesis (NIMBioS), Knoxville, Tennessee, USA
DEB and biodiversity: possible mechanisms
for species coexistence
(1) mutual syntrophy, where the fate of one species is directly linked to that of
another
(2) nutritional `details': The number of substrates is actually large, even if the
number of species is small
(3) social interaction, which means that feeding rate is no longer a function of
food availability only
(4) spatial structure: extinction is typically local only and followed by
immigration from neighbouring patches
(5) temporal structure
Kooijman 2010, chapter 9
Examples of syntrophic symbiosis
2 ORGANISMS
PHOTOAUTOTROPH
HETEROTROPH
2 ORGANS
“PHOTOAUTOTROPH”
“HETEROTROPH”
S.A.L.M. Kooijman, Phil Trans Roy Soc, 2001
SHARING THE SURPLUS
• HOST RECEIVES PHOTOSYNTHATE SYMBIONT CANNOT USE
• SYMBIONT RECEIVES NITROGEN HOST CANNOT USE
Model features and predictions
• Assumes V1 morphs
• Dynamics from set of ODEs coupled to
nonlinear algebraic equations
• Asymptotic “balanced” exponential
growth/decline with stable
host/symbiont ratio
• With plausible parameters,
host/symbiont ratio consistent with
observations on corals
E.B. Muller et al., J. Theor. Biol. 2009
P.J. Edmunds et al. Oecologia, 2011
ABSTRACTIONS IN THREE FANTASY WORLDS
DEB theory
Structure
Reserve
Homestasis
Maturity
Macrochemical
equations
Biomathematics
Steady state
Stability
Cycles
Bifurcations
Optimality
Evolutionary
theory
Fitness
Life history
strategies
Invasibility
ESS
Understanding can advance by coupling ideas from these three worlds
Evolutionary theory of balanced growth
Problem
Given:
dS
 u (t ) f ( S , R );
dt
dR
 (1  u (t )) f ( S , R)
dt
Find u(t) such that biomass is maximized at (large) T > 0
T
 4
Solution
For t < some t0, u(t) = 1 or 0 depending on initial conditions
f f

For t > t0, there is “balanced growth” with
R S
Iwasa and Roughgarden Theor Pop Biol, 1984;
Velten and Richter Bull. Math. Bio. 1995
“Toy model” of sharing the surplus
“Toy model” of sharing the surplus in a plant
Mass Balance
dS
 QS  TS ;
dt
dR
 QR  TR
dt
Uptake
UC  C S ;
U N   N R;
Loss
Ts   S S ;
TR   R R;
N-Recycling
rs   S  S  S S ;
Production
SU Rejection
Waste
rR   R  R R R;
Qs   U C ,  S 1   N  rS   ; QR    C ,  R 1 U N  rR  
c  U C  QS ;
wC  C  QR ;
 N  U N  rR   R QR
wN   N  rS   S QS
2 ODEs + 2 nonlinear algebraic equations
MULTIPLE REPRESENTATIONS OF SU
• k=1 (red);
• k=1.7 (green);
• k=10 (dark blue).
• The curve with gradient
discontinuity is the minimum rule.
• The broken green line is the PCSU
– almost indistinguishable from the
function plotted with k=1.7.
Toy Model Dynamics
Root and Shoot biomasses
• Initially one player supports
growth of the other
• Then “balanced”
exponential growth
Log (biomasses)
root
shoot
Toy Model Dynamics
Root and Shoot biomasses
Log (biomasses)
root
shoot
• Initially one player supports
growth of the other
• Then “balanced”
exponential growth
• Exponential growth rate
satisfies Velten-Richter
condition for optimal growth
rate
Suggests that DEB-inspired concept of “sharing the surplus” is an
evolutionarily stable strategy
Surplus sharing not always “optimal” with minimum rule
Log (biomass)
Root and shoot biomass (log scale)
• Previous runs had
lower C:N ratio in
shoots than roots
• With low-N leaves,
there is oscillatory
growth pattern
(overcompensation)
Dynamics with PCSU
Low C:N shoot
Very high C:N shoot
Evolutionary analysis result
Asymptotic rate of balanced
growth is still optimal in the
sense that no reallocation
of elements by some global
(e.g. hormonal) controller
increases the growth rate.
Oscillatory trajectory is
non-optimal
Application to Lemna (duckweed)
Experiments by Elke Zimmer
0.4
0.35
Root / shoot ratio
0.3
L1
0.25
L2 - a
L2 - b
0.2
L3
0.15
L4
0.1
L5
0.05
L6
0
2 wk low
3 wk low
2 wk high
Nutrient treatment
3 wk high
low -> high
• Experiments with different light and nutrient
(NO3) levels
• NO3 is absorbed by root and shoot, but
there is evidence from literature that there
is translocation to root for reduction. Minor
model change.
• Model always predicts that root/shoot ratio
increases with incident light
• Model parameterization for quantitative
tests in progress
Application to Corals
Work with Ross Cunning
• Toy model elucidates the stabilizing
mechanism for syntophic symbiosis
• Many simple mathematical models show that
combination of negative feedback
(stabilizing) and positive feedback
(destabilizing) can lead to multiple steady
states and hysteresis
• Model elaboration added a positive feedback
with CO2 from host being concentrated and
enhancing photosynthesis
• Model also added photoinhibition and
photodamage
Symbiont/host ratio
Coral model predicts delayed response from bleaching
Two steady states with symbiont Climited or N-limited
After period of high light symbiont
response is delayed
Serious mathematical and numerical
challenges as algebraic equations did not
have unique solution
Figure is projection of higher dimensional
system – so trajectories can cross
Take home message
DEB theory
Biomathematics
Evolutionary theory
Recognizing mutualism among theories accelerates understanding