Metabolism I

contributions from :
Tânia Sousa
Gonçalo Marques and Bas Kooijmann
A Theory of Metabolism
 What is metabolism?

A Theory of Metabolism
 What is metabolism?

 “Using resources (energy and materials) to make new
cells, to repair old ones, and to get rid of wastes
requires the assemblage of biochemical pathways that
we call metabolism. Metabolism is a universal feature
of life that links organisms with their environment,
and with each other.”
A Theory of Metabolism
 What is metabolism?

 “Using resources (energy and materials) to make new
cells, to repair old ones, and to get rid of wastes
requires the assemblage of biochemical pathways that
we call metabolism. Metabolism is a universal feature
of life that links organisms with their environment,
and with each other.”
 What should a theory of metabolism look like?
A Theory of Metabolism

 What is metabolism?
 What should a theory of metabolism look like?
 It should be a qualitative and quantitative description
of how organisms use mass and energy to do the
things they need to do to stay alive
A Theory of Metabolism

 What is metabolism?
 What should a theory of metabolism look like?
 It should be a qualitative and quantitative description
of how organisms use mass and energy to do the
things they need to do to stay alive
 Which type of questions can a theory of metabolism
help you with?
A Theory of Metabolism

 Which type of questions can a theory of metabolism
help you with?
 Why do weak accids have an effect on the growth of
yeasts?
 What is the minimum amount of food (and
habitat) a panda needs to survive?
 If the temperature of ocean increases by 0.5ºC what
will happen to the survival of the sardine larvae?
Environmental Applications
 Toxicology

Environmental Applications
 Toxicology

 Which is the toxicity of the
environmental concentration of a
compound?
 Which are the toxic effects of a
compound?
 Climate Change
Environmental Applications
 Toxicology

 Which is the toxicity of the
environmental concentration of a
compound?
 Which are the toxic effects of a
compound?
 Climate Change
 Will an increase in 1ºC have a drastic
impact on the distribution range of a
species?
 Waste water treatment plant
Environmental Applications
 Toxicology

 Which is the toxicity of the environmental
concentration of a compound?
 Which are the toxic effects of a compound?
 Climate Change
 Will an increase in 1ºC have a drastic impact
on the distribution range of a species?
 Waste water treatment plant
 What are the necessary conditions to mantain
an healthy microbian comunity in the
biological reactors?
 Fisheries Management
Environmental Applications
 Toxicology

 Which is the toxicity of the environmental
concentration of a compound?
 Which are the toxic effects of a compound?
 Climate Change
 Will an increase in 1ºC have a drastic impact
on the distribution range of a species?
 Waste water treatment plant
 What are the necessary conditions to mantain
an healthy microbian comunity in the
biological reactors?
 Fisheries Management
 What is the sustainable fishing quota?
DEB – a theory for Metabolism

 It captures the quantitative aspects of metabolism at
the individual level for all species
DEB – a theory for Metabolism

 It captures the quantitative aspects of metabolism at the
individual level for all species
 Why the hope for generality?
 universality of physics and evolution
 Entropy production is >=0
 cell universality:
 cells are metabolically very similar,
independently of the organism or its size
 widespread biological empirical patterns
A widespread biological empirical fact:
Von Bertalanffy growth

 Growth as a function of time
L(t )  L  ( L  Lb ) e  rB t
 Depends on length at birth,
maximum length and
growth rate
 It was proposed in 1929 by
Putter and in 1938 by Von
Bertalanffy
A widespread biological empirical fact:
Kleiber’s Law

 Metabolism (respiration or
heat production) as a
function of mass
M  aW b
 Metabolism increases with
weight raised to the power
3/4
 Max Kleiber originally
formulated this basic
relationship back in the
1930s.
What is the relationship between
specific metabolism and weight?
A widespread biological empirical fact:
Kleiber’s Law

 Relationship between specific metabolism and weight?
Basic Concepts in DEB Theory

 Consistency with other scientific knowledge
(thermodynamics, evolution, etc)
 Consistency with empirical data
 Life-cycle approach: embryo,
juvenile and adult
 Occam’s razor: the general model should be as
simple as possible (and not more)
 Occam’s biological razor: organisms increased their
control over metabolism during evolution
Basic Concepts in DEB Theory

 The individual: time and spatial scales
A DEB organism: the State Variables

 Metabolism in a DEB
individual.
 The boundary of the
organism
 Rectangles are state
variables
DEB model: the State Variables

 What defines a DEB organism?
 Biomass
 V - Structure
 E - Reserve
 Life-Cycle approach: different life stages
 EH – Maturity level (it doesn’t contribute to the
organisms mass)
 What about other possibles state variables such as
age?
Why not age as a state variable?

Trichopsis vittatus
These gouramis are from the same nest,
they have the same age and lived in the same tank
Social interaction during feeding caused the huge size difference
Age-based models for growth are bound to fail;
growth depends on food intake
Notation
General
Indices for compounds
Indices for transformations

DEB model: Reserve and Structure
 Strong homeostasis

 Reserve & structure have constant aggregated
chemical composition
DEB model: Reserve and Structure
 Strong homeostasis

 Reserve & structure have constant aggregated
chemical composition
 Why more than 1 state variable to define the biomass?
 The aggregated chemical composition of organisms is not constant –
it changes with the growth rate
What does a variable aggregated chemical composition
impliy?
 Why not more than 2 state variables to define biomass?
 Two are sufficient (in animals and bacteria) to capture the change in
aggregated chemical composition with the growth rate
 Strong homeostasis → higher control over metabolism
Why not use thousands of chemical species
and chemical reactions to define the organism?

 Metabolism at the chemical level is very complex
 It is not possible to impose mass conservation without
modeling all chemical reactions (which is impossible).
 “Knowledge on motors of cars is of little help to solve queuing
problems”
DEB model: Reserve and Structure
 Weak homeostasis

 At constant food organisms tend to constant
aggregated chemical composition
DEB model: Reserve and Structure
 Weak homeostasis

 At constant food organisms tend to constant
aggregated chemical composition
What has to be the relationship between V and E to
ensure a constant aggregated chemical composition?
 Empirical support: growing biomass tends to constant
chemical composition at constant food
 Weak homeostasis -> higher control over metabolism
DEB model: Maturity

 Life Stages (dark blue) and transitions (light blue)
embryo
fertilization
juvenile
birth
adult
puberty
death
DEB model: Maturity

 Life Stages (dark blue) and transitions (light blue)
embryo
fertilization
baby
birth
juvenile
adult
infant
weaning
puberty
death
 Essential switch points for metabolic behavior
 Birth (start of feeding)
 Puberty (start of allocation to reproduction)
 Switch points sometimes in reversed order (aphids)
EHb- threshold of maturity at birth
EHp- threshold of maturity at puberty
Life-stages: Metamorphosis

Life-stages: Metamorphosis

Venturia cannescens
Larva
Embryo
Puppa
Adult
EHb - Extremes in relative maturity at birth

Didelphus marsupiales (Am opossum)
♂, ♀ 0.5 + 0.5 m, 6.5 kg
At birth: <2 g; ab = 8-13 d
10-12 (upto 25) young/litter, 2 litters/a
Ommatophoca rossii (Ross Seal)
♂ 1.7-2.1 m, 129-216 kg
♀ 1.3-2.2 m, 159-204 kg
At birth: 1 m, 16.5 kg; ab = 270 d
A DEB organism – energy description

 Metabolism in a DEB
individual.
 The boundary of the
organism
 Rectangles are state
variables
Why is the chemical potential of reserve a parameter?
A DEB organism – energy description

 Metabolism in a DEB
individual.
 The boundary of the
organism
 Rectangles are state
variables
 Chemical and thermodynamic properties of the structure and
𝐸 = 𝑀𝐸 𝜇𝐸
reserve are constant (strong homeostasis)
𝜇𝐸 - chemical potential of reserve
𝜇𝑉 - chemical potential of structure
EHb- threshold of maturity at birth
EHp- threshold of maturity at puberty
𝐸𝐻 = 𝑀𝐻 𝜇𝐸
𝐸𝑉 = 𝑀𝑉 𝜇𝑉
MHb- threshold of maturity at birth
MHp- threshold of maturity at puberty
A DEB organism

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food pX,
reserve pA, pC, pS, pT , pG,
pR, pJ or structure pVG.
 Circles are transformations
Notation
General
Indices for compounds
Indices for transformations

Notation for mass flows 1

Feeding & Assimilation

2/3
𝑝𝑋 = 𝑓(𝑋) 𝑝𝑋𝑚 𝑉
 Feeding: the uptake of food
 Assimilation: conversion of substrate (food,
nutrients, light) into reserve(s) 𝑝𝐴 = 𝜅𝑋 𝑝𝑋 = 𝑓(𝑋) 𝑝𝐴𝑚 𝑉 2/3
 Depend on substrate availability & structural surface
area (e.g. surface area of the gut)
𝑝𝑋𝑚 - surface maximum feeding rate
𝑝𝐴𝑚 - surface maximum assimilation rate
𝜅𝑋 - efficiency of food assimilation
Are all these
parameters
independent?
Feeding & Assimilation

2/3
𝑝𝑋 = 𝑓(𝑋) 𝑝𝑋𝑚 𝑉
 Feeding: the uptake of food
 Assimilation: conversion of substrate (food,
nutrients, light) into reserve(s) 𝑝𝐴 = 𝜅𝑋 𝑝𝑋 = 𝑓(𝑋) 𝑝𝐴𝑚 𝑉 2/3
 Depend on substrate availability & structural surface
area (e.g. surface area of the gut)
𝑝𝑋𝑚 - surface maximum feeding rate
𝑝𝐴𝑚 - surface maximum assimilation rate
𝜅𝑋 - efficiency of food assimilation
Are all these
parameters
independent?
 Empirical pattern: the heat increment of feeding suggests that
there are processes only associated with food processing
 Consistency with other fields: mass transfer (needed for
acquisition, digestion and food processing) is proportional to
area
Why is the efficiency of food assimilation of reserve on food a parameter?
Feeding & Assimilation

2/3
𝑝𝑋 = 𝑓(𝑋) 𝑝𝑋𝑚 𝑉
 Feeding: the uptake of food
 Assimilation: conversion of substrate (food,
nutrients, light) into reserve(s) 𝑝𝐴 = 𝜅𝑋 𝑝𝑋 = 𝑓(𝑋) 𝑝𝐴𝑚 𝑉 2/3
 Depend on substrate availability & structural surface
area (e.g. surface area of the gut)
𝑝𝑋𝑚 - surface maximum feeding rate
𝑝𝐴𝑚 - surface maximum assimilation rate
𝜅𝑋 - efficiency of food assimilation
 Empirical pattern: the heat increment of feeding suggests that
there are processes only associated with food processing
 Strong homeostasis imposes a fixed conversion efficiency
 Consistency with other fields: mass transfer (needed for
acquisition, digestion and food processing) is proportional to
area
Feeding rate

Filtration rate, l/h
 If food availability is constant (or abundant) feeding
increases proportional to area or L2 (for isomorphs)
Mytilus edulis
Data: Winter 1973
Length, cm
Intra-taxon predation: efficient conversion
κX a high efficiency of food assimilation

Hemiphractus fasciatus
is a frog-eating frog
Beroe sp
is a comb jelly-eating comb jelly
Euspira catena
is a snail-eating snail
Coluber constrictor
is a snake-eating snake
Solaster papposus
is a starfish-eating starfish
Chrysaora hysoscella
is a jelly fish-eating jelly fish
Intra-taxon predation: efficient conversion
κX a high efficiency of food assimilation

Asplanchna girodi
is a rotifer-eating rotifer
Falco peregrinus
is a bird-eating bird
Didinium nasutum
is a ciliate-eating ciliate
Acinonyx jubatus
is a mammal-eating mammal
Esox lucius
is a fish-eating fish
Enallagma carunculatum
is a insect-eating insect
Feeding & Assimilation

 Proportionality between
assimilation and ingestion
rate for Mytilus Edulis for
different sizes
A DEB organism

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food pX,
reserve pA, pC, pS, pT , pG,
pR, pJ or structure pVG.
 Circles are transformations
𝑝𝑋 = 𝑓(𝑋) 𝑝𝑋𝑚 𝑉 2/3
How do we obtain the mass description?
𝑝𝐴 = 𝜅𝑋 𝑝𝑋 = 𝑓(𝑋) 𝑝𝐴𝑚 𝑉 2/3
𝑝𝑋𝑚 - surface maximum feeding rate
𝑝𝐴𝑚 - surface maximum assimilation rate
𝜅𝑋 -efficiency of food assimilation
A DEB organism – energy description

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food
JXA, reserve JEA, JEC, JEM, JET ,
JEG, JER, JEJ or structure JVG.
 Circles are transformations
𝐽𝑋𝐴 = 𝑓(𝑋) 𝐽𝑋𝐴𝑚 𝑉 2/3
𝑝𝑋 = 𝐽𝑋𝐴 𝜇𝑋 = 𝑓(𝑋) 𝑝𝑋𝑚 𝑉 2/3
𝐽𝐸𝐴 = 𝑦𝐸𝑋 𝐽𝑋𝐴 = 𝑓(𝑋) 𝐽𝐸𝐴𝑚 𝑉 2/3
𝑝𝐴 = 𝐽𝐸𝐴 𝜇𝐸 = 𝑓(𝑋) 𝑝𝐴𝑚 𝑉 2/3
𝐽𝑋𝐴𝑚 - surface maximum feeding rate
𝐽𝐸𝐴𝑚 - surface maximum assimilation rate
𝑦𝐸𝑋 - yield of reserve on food
𝑝𝑋𝑚 = 𝜇𝑋 𝐽𝑋𝐴𝑚
𝑝𝐴𝑚 = 𝜇𝐸 𝐽𝐸𝐴𝑚
Notation
General
Indices for compounds
Indices for transformations

A DEB organism – parameters

Primary parameters
Auxiliary parameters
𝜅𝑋
𝜇𝑋 , 𝜇𝐸 , 𝜇𝑉
𝑝𝐴𝑚
𝑑𝑋 , 𝑑𝐸 , 𝑑𝑉
𝑤𝑋 , 𝑤𝐸 , 𝑤𝑉
A DEB organism
Mobilization

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food pX,
reserve pA, pC, pS, pT , pG,
pR, pJ or structure pVG.
 Circles are transformations
Mobilization of Reserve

 The mobilization of reserve is used to fuel the
organism’s activities
𝑝𝐶 = 𝑝𝑆 +𝑝𝐺 +𝑝𝐽 + 𝑝𝑅
𝑣
𝐸
𝑑𝐿
𝑝𝐶 = 𝐸
−𝑟 =
𝑣−3
𝐿
𝐿
𝑑𝑡
𝑣 - energy conductance
Mobilization of Reserve

 The mobilization of reserve is used to fuel the
organism’s activities
𝑝𝐶 = 𝑝𝑆 +𝑝𝐺 +𝑝𝐽 + 𝑝𝑅
𝑣
𝐸
𝑑𝐿
𝑝𝐶 = 𝐸
−𝑟 =
𝑣−3
𝐿
𝐿
𝑑𝑡
𝑣 - energy conductance






Empirical pattern: organisms are capable of spending energy on growth, maintenance
and reproduction in the absence of food
Empirical pattern: eggs spend energy on maturation but do not feed.
Mobilization from reserve -> higher control over the metabolism (independence from
the environment)
Mobilization is uncoupled from assimilation & feeding: makes evolution easier
Weak homeostasis & strong homeostasis & independence of mobilization from the
environment  mobilization is proportional to C-moles of reserve per unit of length
What are the units of 𝑣?
Notation
General
Indices for compounds
Indices for transformations

Notation 3

Notice that some symbols have more than one meaning:
V as symbol stands for volume, and without index for volume of structure,
as index stands for the compound structure
E as symbol stands for energy, and without index for energy in reserve,
as index stands for the compound reserve
C, H, O, N as indices stand for mineral compounds as well as chemical
elements the context defines the meaning
Dots are used to
• distinguish rates from states (dimension check)
• allow scaling of time without the need to introduce new symbols
if time is scaled to a dimensionless quantity, the dot is removed
Mobilization of Reserve

Some populations of humpback whale Megaptera novaeangliae (36 Mg)
migrate 26 Mm anually without feeding,
A 15 m mother gets a 6 m calf in tropical waters, gives it 600 l milk/d for 6 months and
together return to cold waters to resume feeding in summer
A DEB organism – energy description

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food pX,
reserve pA, pC, pS, pT , pG,
pR, pJ or structure pVG.
 Circles are transformations
𝑝𝐶 = 𝑝𝑆 +𝑝𝐺 +𝑝𝐽 +𝑝𝑅
𝑝𝐶 = 𝐸
𝑣
−𝑟
𝐿
1 𝑑𝑉
𝑟=
𝑉 𝑑𝑡
Write in the
mass framework
A DEB organism – energy description

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food pX,
reserve pA, pC, pS, pT , pG,
pR, pJ or structure pVG.
 Circles are transformations
𝑝𝐶 = 𝑝𝑆 +𝑝𝐺 +𝑝𝐽 +𝑝𝑅
𝑝𝐶 = 𝐸
𝑣
−𝑟
𝐿
1 𝑑𝑉
𝑟=
𝑉 𝑑𝑡
Write in the
mass framework
𝐽𝐸𝐶 = 𝐽𝐸𝑆 +𝐽𝐸𝐺 +𝐽𝐸𝐽 + 𝐽𝐸𝑅
𝐽𝐸𝐶 = 𝑀𝐸
𝑣
−𝑟
𝐿
𝑟=
1 𝑑𝑀𝑉
𝑀𝑉 𝑑𝑡
A DEB organism
The kappa rule – a fixed allocation rule

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food pX,
reserve pA, pC, pS, pT , pG,
pR, pJ or structure pVG.
 Circles are transformations
𝜅 rule

 A fixed fraction  of mobilised reserve is allocated to
somatic maintenance and growth, the rest to maturity
maintenance and maturation (juveniles) or reproduction
(1 − κ)𝑝𝐶 = 𝑝𝐽 +𝑝𝑅
(adults). κ𝑝𝐶 = 𝑝𝑆 +𝑝𝐺
 - kappa
 Empirical pattern: some species do not stop growing after
reproduction has started
Length, mm
Cum # of young
Reproduction 
Growth:
d
L  rB ( L  L)
dt
Von Bertalanffy
Age, d
Age, d
𝜅 rule

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food pX,
reserve pA, pC, pS, pT , pG,
pR, pJ or structure pVG.
 Circles are transformation
 The kappa rule (a fixed
allocation rule)
𝜅𝑝𝐶 = 𝑝𝑆 + 𝑝𝐺
𝜅𝐽𝐸𝐶 = 𝐽𝐸𝑆 +𝐽𝐸𝐺
(1 − 𝜅) 𝑝𝐶 = 𝑝𝐽 +𝑝𝑅
(1 − 𝜅)𝐽𝐸𝐶 = 𝐽𝐸𝐽 +𝐽𝐸𝑅
A DEB organism – parameters

Primary parameters
Auxiliary parameters
𝜅𝑋
𝜇𝑋 , 𝜇𝐸 , 𝜇𝑉
𝑝𝐴𝑚
𝑑𝑋 , 𝑑𝐸 , 𝑑𝑉
𝑣
𝜅
𝑤𝑋 , 𝑤𝐸 , 𝑤𝑉
A DEB organism
Priority allocation rules

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food pX,
reserve pA, pC, pS, pT , pG,
pR, pJ or structure pVG.
 Circles are transformations
 The kappa rule (a fixed
allocation rule)
 The priority maintenance
rules
The priority maintenance rule states that maintenance has priority:
from κ 𝑝𝐶 = 𝑝𝑆 +𝑝𝐺 somatic maintenance is paid first and the rest
goes to growth
A DEB organism
Priority allocation rules

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food pX,
reserve pA, pC, pS, pT , pG,
pR, pJ or structure pVG.
 Circles are transformations
 The kappa rule (a fixed
allocation rule)
 The priority maintenance
rules
The priority maintenance rule states that maintenance has priority:
from κ 𝑝𝐶 = 𝑝𝑆 +𝑝𝐺 somatic maintenance is paid first and the rest
goes to growth
A DEB organism – somatic maintenance

 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food pX,
reserve pA, pC, pS, pT , pG,
pR, pJ or structure pVG.
 Circles are transformations
 The kappa rule (a fixed
allocation rule)
 The priority maintenance
rules
Somatic maintenance

 Collection of processes that maintain the organism
alive:
 protein turnover (synthesis, but no net synthesis)
 maintaining conc. gradients across membranes
(proton leak)
 (some) product formation (leaves, hairs, skin flakes,
moults)
 movement (usually less than 10% of maintenance
costs)
Somatic maintenance

 Reserve compounds have no maintenance needs
because they have a limited lifetime
 Somatic maintenance is paid from flux  pC:
𝑝𝑆 = 𝑝𝑀 𝐿3 + 𝑝𝑇 𝐿2 = 𝑝𝑀 𝑉 + 𝑝𝑇 𝑉 2/3
  structural volume (most costs)
  surface area: heating (endotherms), osmo-regulation
(fresh water organisms)
 Specific somatic maintenance costs are constant because the
chemical and thermodynamic properties of the structure are
constant (strong homeostasis)
 Empirical pattern: Freshly produced eggs consist primarly of
reserve and hardly respire
𝑝𝑀 - volume specific maintenance costs
𝑝𝑇 - surface specific maintenance costs
Reserve pays no maintenance
embryonic development

embryo
weight, g
yolk
time, d
O2 consumption, ml/h
Carettochelys insculpta
Data from Web et al 1986
time, d
A DEB organism – parameters

Primary parameters
Auxiliary parameters
𝜅𝑋
𝜇𝑋 , 𝜇𝐸 , 𝜇𝑉
𝑝𝐴𝑚
𝑑𝑋 , 𝑑𝐸 , 𝑑𝑉
𝑣
𝜅
𝑤𝑋 , 𝑤𝐸 , 𝑤𝑉
𝑝𝑀
𝑝𝑇