Physical & Chemical Constraints on Population Dynamics

Bas KooijmanDept of Theoretical Biology

Vrije Universiteit, Amsterdamhttp://www.bio.vu.nl/thb/deb/

Leiden, 2004/12/10Hans Metz 60th birthday

adul

t

embryo

juvenile

25 year research onDynamic Energy Budget

theory for metabolic organisation

DEB – ontogeny - IBM1980

1990

2000

Daphniaecotox

application

NECs

ISO/OECD

embryos

body sizescaling

morphdynamicsindirect

calorimetry

food chains

SynthesizingUnits

multivarplants

adaptationtumour

induction

von Foerster

epidemiolapplications

bifurcationanalysis

Globalbif-analysis

integralformulations

adaptive dynamics

ecosystem Self-orginazation

numericalmethods

symbioses

ecosystemdynamics

molecularorganisation

DEB 1

DEB 2

DEBtox 1

organfunction

aging

micro’s

molecule

cell

individual

population

ecosystem

system earth

time

spac

e

Space-time scales

When changing the space-time scale, new processes will become important other will become less importantIndividuals are special because of straightforward energy/mass balances

Each process has its characteristic domain of space-time scales

Empirical special cases of DEB year author model year author model

1780 Lavoisier multiple regression of heat against mineral fluxes

1950 Emerson cube root growth of bacterial colonies

1825 Gompertz Survival probability for aging 1951 Huggett & Widdas foetal growth

1889 Arrhenius temperature dependence of physiological rates

1951 Weibull survival probability for aging

1891 Huxley allometric growth of body parts 1955 Best diffusion limitation of uptake

1902 Henri Michaelis--Menten kinetics 1957 Smith embryonic respiration

1905 Blackman bilinear functional response 1959 Leudeking & Piret microbial product formation

1910 Hill Cooperative binding 1959 Holling hyperbolic functional response

1920 Pütter von Bertalanffy growth of individuals

1962 Marr & Pirt maintenance in yields of biomass

1927 Pearl logistic population growth 1973 Droop reserve (cell quota) dynamics

1928 Fisher & Tippitt

Weibull aging 1974 Rahn & Ar water loss in bird eggs

1932 Kleiber respiration scales with body weight3/ 4

1975 Hungate digestion

1932 Mayneord cube root growth of tumours 1977 Beer & Anderson development of salmonid embryos

DEB theory is axiomatic, based on mechanisms not meant to glue empirical models

Since many empirical models turn out to be special cases of DEB theory the data behind these models support DEB theory

This makes DEB theory very well tested against data

Some DEB pillars• life cycle perspective of individual as primary target embryo, juvenile, adult (levels in metabolic organization)

• life as coupled chemical transformations (reserve & structure)

• time, energy, entropy & mass balances

• surface area/ volume relationships (spatial structure & transport)

• homeostasis (stoichiometric constraints via Synthesizing Units)

• syntrophy (basis for symbioses, evolutionary perspective)

• intensive/extensive parameters: body size scaling

1- maturitymaintenance

maturityoffspring

maturationreproduction

Basic DEB scheme

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

1-

1-u

Competitive tumour growth

food faecesassimilation

reserve

feeding defecation

structurestructure

somaticmaintenance

growth

maturitymaintenance

maturityoffspring

maturationreproduction

tumourtumour

u

)(][)(][

)(][)(

tVptVp

tVptκ

uMuM

uMuu

Allocation to tumour relative maint workload

Isomorphy: is constantTumour tissue: low spec growth costs low spec maint costs

uκ

Van Leeuwen et al., 2003 The embedded tumour: host physiology is important for the evaluation of tumour growth.British J Cancer 89, 2254-2268

maint

Biomass: reserve(s) + structure(s)

Reserve(s), structure(s): generalized compounds, mixtures of proteins, lipids, carbohydrates: fixed compositionCompounds in reserve(s): equal turnover times, no maintenance costs structure: unequal turnover times, maintenance costs

Reasons to delineate reserve, distinct from structure• metabolic memory• explanation of respiration patterns (freshly laid eggs don’t respire) • biomass composition depends on growth rate• fluxes are linear sums of assimilation, dissipation and growth basis of method of indirect calorimetry• explanation of inter-species body size scaling relationships

Biomass compositionData Esener et al 1982, 1983; Kleibsiella on glycerol at 35°C

nHW

nOW

nNW

O2

CO2Spec growth rate, h-1

Spec growth rate

Spec growth rate, h-1

Rel

ativ

e ab

unda

nce

Spe

c pr

od, m

ol.m

ol-1.h

-1

Wei

ght y

ield

, mol

.mol

-1

nHE 1.66 nOE 0.422 nNE 0.312nHV 1.64 nOV 0.379 nNV 0.189

kE 2.11 h-1 kM 0.021 h-1

yEV 1.135 yXE 1.490rm 1.05 h-1 g = 1

•μE-1 pA pM pG

JC 0.14 1.00 -0.49

JH 1.15 0.36 -0.42

JO -0.35 -0.97 0.63

JN -0.31 0.31 0.02

Entropy J/C-mol.K Glycerol 69.7 Reserve 74.9 Structure 52.0

Sousa et al 2004Interface, subm

Yield vs growth

1/spec growth rate, 1/h

1/yi

eld,

mm

ol g

luco

se/

mg

cells

Streptococcus bovis, Russell & Baldwin (1979)

Marr-Pirt (no reserve)DEB

spec growth rate

yield

Russell & Cook (1995): this is evidence for down-regulation of maintenance at low growth ratesDEB theory: high reserve density gives high growth rates structure requires maintenance, reserves not

Inter-species body size scaling• parameter values tend to co-vary across species• parameters are either intensive or extensive• ratios of extensive parameters are intensive• maximum body length is allocation fraction to growth + maint. (intensive) volume-specific maintenance power (intensive) surface area-specific assimilation power (extensive)• conclusion : (so are all extensive parameters)• write physiological property as function of parameters (including maximum body weight)• evaluate this property as function of max body weight

][/}{ MAm pκpL

}{ Ap][ Mp

mA Lp }{

Kooijman 1986Energy budgets can explain body size scaling relationsJ. Theor. Biol. 121: 269-282

Scaling of metabolic rate

comparison intra-species inter-species

maintenance

growth

weight

nrespiratio3

32

dl

llls

43

32

ldld

lll

EV

h

structure

reserve

32 lll

l0l

0

3lllh

Respiration: contributions from growth and maintenanceWeight: contributions from structure and reserveStructure ; = length; endotherms 3l l

3lllh

0hl

Metabolic rate

Log weight, g

Log metabolic rate,

w

endotherms

ectotherms

unicellulars

slope = 1

slope = 2/3

Length, cm

O2 consum

ption,

l/h

Inter-speciesIntra-species

0.0226 L2 + 0.0185 L3

0.0516 L2.44

2 curves fitted:

(Daphnia pulex)

Synthesizing Unit dynamicsSU: Generalized enzyme that operates on fluxes of metabolites

Typical form for changes in bounded fractions

Typical flux of metabolites for

Mixing of types:

Example of mixture between sequential & complementary substrates:

Interactions of substrates

Kooijman, 2001Phil Trans R Soc B356: 331-349

Co-metabolismCo-metabolic degradation of 3-chloroaniline by Rhodococcus with glucose as primary substrateData from Schukat et al, 1983

Brandt et al, 2003Water Research37, 4843-4854

Aggressive competition

V structure; E reserve; M maintenance substrate priority E M; posteriority V MJE flux mobilized from reserve specified by DEB theoryJV flux mobilized from structure amount of structure (part of maint.) excess returns to structurekV dissociation rate SU-V complex kE dissociation rate SU-E complex kV kE depend on such that kM = yMEkE(E. + EV)+yMVkV .V is constant

J EM,

J VM

J EM,

J VM

JE

kV = kE

kV < kE

Collaboration:Tolla, Poggiale, Auger, Kooi, Kooijman

Behaviour Energetics

DEB fouraging module: time budgeting

• Fouraging feeding + food processing, food selection

feeding surface area (intra-species), volume (inter-species)

• Sleeping repair of damage by free radicals respiration

respiration scales between surface area & volume

• Social interaction feeding efficiency (schooling)

resource partitioning (territory) mate selection (gene quality energetic parameter values)

• Migration traveling speed and distance: body size spatial pattern in resource dynamics (seasonal effects) environmental constraints on reproduction

Social inhibition of x esequential parallel

dilution rate

subs

trat

e co

nc.

biom

ass

conc

.

No

soci

aliz

atio

n

Implications: stable co-existence of competing species “survival of the fittest”? absence of paradox of enrichment

x substratee reservey species 1z species 2

Collaboration:Van Voorn, Gross, Feudel, Kooi, Kooijman

Significance of co-existence

Main driving force behind evolution:• Darwin: Survival of the fittest (internal forces) involves out-competition argument• Wallace: Selection by environment (external forces) consistent with observed biodiversity

Mean life span of typical species: 5 - 10 Ma

Sub-optimal rare species: not going extinct soon (“sleeping pool of potential response”) environmental changes can turn rare into abundant species

Surface area/volume interactions 2.2

• biosphere: thin skin wrapping the earth light from outside, nutrient exchange from inside is across surfaces production (nutrient concentration) volume of environment• food availability for cows: amount of grass per surface area environ food availability for daphnids: amount of algae per volume environ• feeding rate surface area; maintenance rate volume (Wallace, 1865)

• many enzymes are only active if linked to membranes (surfaces) substrate and product concentrations linked to volumes change in their concentrations gives local info about cell size; ratio of volume and surface area gives a length

Change in body shapeIsomorph: surface area volume2/3

volumetric length = volume1/3

V0-morph: surface area volume0

V1-morph: surface area volume1

Ceratium

Mucor

Merismopedia

Shape correction functionShape correction function

at volume Vactual surface area at volume V

isomorphic surface area at volume V=

1)( VΜ for dVV

V0-morphV1-morph isomorph 0

3/1

3/2

)/()(

)/()(

)/()(

d

d

d

VVV

VVV

VVV

Μ

Μ

Μ

3/13/2

3/13/2

)/(2

2)/(

2)(

)/(3

3)/(

3)(

dd

dd

VVδ

VVδ

δV

VVδ

VVδ

V

Μ

Μ

Static mixtures between V0- and V1-morphs for aspect ratio δ

Mixtures of changes in shapeDynamic mixtures between morphs

Lichen Rhizocarpon

V1- V0-morph

V1- iso- V0-morph

outer annulus behaves as a V1-morph, inner part as a V0-morph. Result: diameter increases time

Biofilms

Isomorph: V1 = 0

V0-morph: V1 =

mixture between iso- & V0-morph

biomass grows, butsurface area that is involvedin nutrient exchange does not

solid substratebiomass

3/2

1

1)(

d

d

VV

VV

V

VVΜ

Size-structured Unstructured Population Dynamics

Isomorphs: individual-based or pde formulationV1-morphs: unstructured (ode) formulation

Effect of individuality becomes small if ratio between largest and smallest body size reduces

This suggest a perturbation method to approximate a pde with an ode formulation

Need for simplification of ecosystem dynamics

Cells, individuals, colonies

• plasmodesmata connect cytoplasm; cells form a symplast: plants

• pits and large pores connect cytoplasm: fungi, rhodophytes

• multinucleated cells occur; individuals can be unicellular: fungi, Eumycetozoa, Myxozoa, ciliates, Xenophyophores, Actinophryids, Biomyxa, diplomonads, Gymnosphaerida, haplosporids, Microsporidia, nephridiophagids, Nucleariidae, plasmodiophorids, Pseudospora, Xanthophyta (e.g. Vaucheria), most classes of Chlorophyta (Chlorophyceae, Ulvophyceae, Charophyceae (in mature cells) and all Cladophoryceae, Bryopsidophyceae and Dasycladophyceae)) • cells inside cells: Paramyxea • uni- and multicellular stages: multicellular spores in unicellular myxozoa, gametes• individuals can remain connected after vegetative propagation: plants, corals, bryozoans• individuals in colonies can strongly interact and specialize for particular tasks: syphonophorans, insects, mole rats

vague boundaries

Kooijman, Hengeveld 2004 The symbiontic nature of metabolic evolution In: Reydon, Hemerik (eds) Current themes in theor biol. Springer, Dordrecht

rotiferConochilus hippocrepisHeterocephalus glaber

Trophic interactions

• Competition for same resources size/age-dependent diet choices

• Syntrophy on products faeces, leaves, dead biomass

• Parasitism (typically small, relative to host) biotrophy, milking, sometimes lethal (disease) interaction with immune system

• Predation (typical large, relative to prey) living individuals, preference for dead/weak specialization on particular life stages (eggs, juveniles) inducible defense systems; cannibalism

Tra

nsit

ions

bet

wee

n th

ese

type

s fr

eque

ntly

occ

ur

Symbiosis

product

substrate

Symbiosis

substrate substrate

Internalization

Structures merge Reserves merge

Free-living, clusteringFree-living, homogeneous

Steps in symbiogenesis 1 substrate+ 1 product

taken up each

2 substrates taken upproducts degradeto physiol role

Symbiogenesis• symbioses: fundamental organization of life based on syntrophy ranges from weak to strong interactions; basis of biodiversity• symbiogenesis: evolution of eukaryotes (mitochondria, plastids)• DEB model is closed under symbiogenesis: it is possible to model symbiogenesis of two initially independently living populations that follow the DEB rules by incremental changes of parameter values such that a single population emerges that again follows the DEB rules• essential property for models that apply to all organisms

Kooijman, Auger, Poggiale, Kooi 2003 Quantitative steps in symbiogenesis and the evolution of homeostasisBiological Reviews 78: 435 - 463

Resource dynamicsTypical approach

Usual form for densities prey x and predator y:

Problems:• Not clear how dynamics depends on properties of individuals, which change during life cycle• If i(x) depends on x: no conservation of mass; popular: i(x) x(1-x/K)• If yield Y is constant: no maintenance, no realism• If feeding function f(cx,cy) cf(x,y) and/or input function i(cx) ci(x) and/or output function o(cx) co(x) for any c>0: no spatial scaling (amount density)Conclusions:• include inert zero-th trophic level (substitutable by mass conservation)• need for mechanistic individual-based population models

Prey/predator dynamics

)(),(

),()(

yoyxfYydt

d

yxfxixdt

d

Kooi et al 1997 J. Biol. Systems, 1: 77-85

Nutrient

Resource dynamics

Resource dynamics

Nutrient

Resource dynamics

Nutrient

Effects of parasites

On individuals: Many parasites

• increase (chemical manipulation)

• harvest (all) allocation to dev./reprod.

Results

• larger body size higher food intake

• reduced reproduction

On populations: Many small parasites

• convert healthy (susceptible) individuals to affected ones on contact

• convert affected individuals into non-susceptible ones

Globif project NWO-CLS programVan Voorn, Kooi, Kooijman

Producer/consumer dynamics

PnCnNPm

ChrCdt

d

CjPrPdt

d

NPNCN

C

PAP

)(

PK

jj

my

kr PAm

PANNP

NP /1

;1

CNCPCNCPC rrrrr

1111

MNPANCNCNMPPACPCP kjmyrkjyr ;

producer

consumer

nutr reserveof producer

: total nutrient in closed system

N

h: hazard rate

CPCCN rry special case: consumer is not nutrient limited

spec growthof consumer

Kooijman et al 2004 Ecology, 85, 1230-1243

Producer/consumer dynamicsConsumer nutrient limited

Consumer notnutrient limited

Hopf bifurcation

Hopf bifurcation

tangent bifurcation

transcritical bifurcation

homoclinicbifurcation

1-species mixotroph community

Mixotrophs areproducers, which live off light and nutrientsas well asdecomposers, which live off organic compounds which they produce by aging

Simplest community with full material cycling

Kooijman, Dijkstra, Kooi 2002 J. Theor. Biol. 214: 233-254

Canonical communityShort time scale:Mass recycling in a community closed for mass open for energy

Long time scale:Nutrients leaks and influxes

Memory is controlled by life span (links to body size)Spatial coherence is controlled by transport (links to body size)

Kooijman, Nisbet 2000 How light and nutrients affect life in a closed bottle. In: Jørgensen, S. E (ed) Thermodynamics and ecological modelling. CRC, 19-60

Self organisation of ecosystems• homogeneous environment, closed for mass • start from mono-species community of mixotrophs• parameters constant for each individual• allow incremental deviations across generations link extensive parameters (body size segregation) • study speciation using adaptive dynamics• allow cannibalism/carnivory• study trophic food web/piramid: coupling of structure & function• study co-evolution of life, geochemical dynamics , climate• adaptive dynamics applied to multi-character DEB models

Troost et al 2004 Math Biosci, to appear; Troost et al 2004 Am Nat, submittedCollaboration: Metz, Troost, Kooi, Kooijman

DEB tele-course 2005

Feb – April 2005, 10 weeks, 200 h no financial costs

http://www.bio.vu.nl/thb/deb/course/