multivariate deb models generalization of the standard deb model with multiple state variables
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Instituto Superior TécnicoDepartamento de Engenharia Mecânica - Secção de Energia e Ambiente 7th DEB Course, 15-23 April 2013
Multivariate DEB modelsGeneralization of the standard DEB model with multiple state variables
Gonçalo Marques
Instituto Superior Técnico,goncalo.marques@ist.utl.pt
7th DEB Course
TEXEL, 15-23 April
Summary
2The standard DEB model is built with one reserve, one structure, one maturity and one reproduction buffer for an organism that feeds on one substrate. There are situations when the standard DEB model is not enough to realistically simulate an organism or a specific feature of an organism. However DEB theory gives us the tools to build models beyond the standard DEB model. The construction of the generalized DEB models will be the theme of this presentation.
1. State variables2. Multiple substrates3. Multiple reserves4. Multiple structures5. Does it make sense to talk about multiple maturities?6. Multiple products
7th DEB Course, 15-23 April 2013
State variables
3Substrate – compound assimilated by an organism
7th DEB Course, 15-23 April 2013
State variables
4Substrate – compound assimilated by an organismReserve – assimilation is stocked in the reserve. The output of the reserve (mobilization flux) will be used for every metabolic purpose.
7th DEB Course, 15-23 April 2013
State variables
5Substrate – compound assimilated by an organismReserve – assimilation is stocked in the reserve. The output of the reserve (mobilization flux) will be used for every metabolic purpose.
7th DEB Course, 15-23 April 2013
State variables
6Substrate – compound assimilated by an organismReserve – assimilation is stocked in the reserve. The output of the reserve (mobilization flux) will be used for every metabolic purpose.
7th DEB Course, 15-23 April 2013
State variables
7Substrate – compound assimilated by an organismReserve – assimilation is stocked in the reserve. The output of the reserve (mobilization flux) will be used for every metabolic purpose.Structure – it is linked to size, and therefore it rules assimilation. It needs to be maintained.
7th DEB Course, 15-23 April 2013
State variables
8Substrate – compound assimilated by an organismReserve – assimilation is stocked in the reserve. The output of the reserve (mobilization flux) will be used for every metabolic purpose.Structure – it is linked to size, and therefore it rules assimilation. It needs to be maintained.
7th DEB Course, 15-23 April 2013
State variables
9Substrate – compound assimilated by an organismReserve – assimilation is stocked in the reserve. The output of the reserve (mobilization flux) will be used for every metabolic purpose.Structure – it is linked to size, and therefore it rules assimilation. It needs to be maintained. Maturity – a measure of thecomplexity of the organism. It also needs to be maintained.
7th DEB Course, 15-23 April 2013
State variables
10Substrate – compound assimilated by an organismReserve – assimilation is stocked in the reserve. The output of the reserve (mobilization flux) will be used for every metabolic purpose.Structure – it is linked to size, and therefore it rules assimilation. It needs to be maintained. Maturity – a measure of thecomplexity of the organism. It also needs to be maintained.
7th DEB Course, 15-23 April 2013
State variables
11Substrate – compound assimilated by an organismReserve – assimilation is stocked in the reserve. The output of the reserve (mobilization flux) will be used for every metabolic purpose.Structure – it is linked to size, and therefore it rules assimilation. It needs to be maintained. Maturity – a measure of thecomplexity of the organism. It also needs to be maintained.Product – compound produced by the organism. It could be released into the environment orremain attached to the organism.Either way it doesn’t needmaintenance.
7th DEB Course, 15-23 April 2013
State variables
12State variables define the state of the system.
In the standard DEB model the system organism-environment is defined by:Organism – reserve, structure, maturity, reproduction bufferEnvironment – substrate
7th DEB Course, 15-23 April 2013
State variables
13Recall one of the important principles of modelling:
Efficiency
Balance the effort and level of detail with the insights your data can provide.
7th DEB Course, 15-23 April 2013
Modelling criteria• Consistency dimensions, conservation laws, realism (consistency with data)• Coherence consistency with neighbouring fields of interest, levels of
organisation• Efficiency comparable level of detail, all vars and pars are effective numerical behaviour• Testability amount of support, hidden variables
7th DEB Course, 15-23 April 2013
Multiple state variables
15
When you need to add state variables remember to ask these questions:
-Does it comply with the DEB core? (coherence)-In what situation do I recover the standard DEB model? And does it make sense? (coherence)-Is this the minimum number of state variables I need to model the feature/behavior/metabolism I want? (efficiency and testability)
𝐸 𝑉𝑋 �̇�𝐴�̇�𝐷 �̇�𝐺
7th DEB Course, 15-23 April 2013
Multiple substrates
16 𝐸 𝑉𝑋
𝑋
One substrate:
𝑋 b
𝑋 p
𝑋 �̇�𝐴�̇�𝑋
7th DEB Course, 15-23 April 2013
�̇�𝑋
�̇�𝐴
𝑋
Multiple substrates
17 𝐸 𝑉𝑋
𝑋
One substrate:
7th DEB Course, 15-23 April 2013
�̇�𝐴 𝑋→𝑃
Multiple substrates
18 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝑋 1+ 𝑋 2→𝑃
complementary
Multiple substrates
19 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
Multiple substrates
20 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
𝜃1 ∙
𝑋 1
Multiple substrates
21 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
𝜃12
𝜃1 ∙
𝑋 1
𝑋 2
Multiple substrates
22 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
𝜃12
𝜃1 ∙
𝑋 1
𝑋 2
𝑃
Multiple substrates
23 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
𝑑𝜃∙ ∙
𝑑𝑡 =¿
𝑑𝜃1 ∙
𝑑𝑡 =¿
𝑑𝜃12
𝑑𝑡 =¿
𝜃12
𝜃1 ∙
Multiple substrates
24 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
𝜃1 ∙
𝑋 1
𝑑𝜃∙ ∙
𝑑𝑡 =− �̇�1 𝑋 1𝜃∙ ∙
𝑑𝜃1 ∙
𝑑𝑡 =�̇�1 𝑋 1𝜃 ∙∙
𝑑𝜃12
𝑑𝑡 =¿
𝜃12
Multiple substrates
25 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
𝜃12
𝜃1 ∙
𝑋 1
𝑋 2
𝑑𝜃∙ ∙
𝑑𝑡 =− �̇�1 𝑋 1𝜃∙ ∙
𝑑𝜃1 ∙
𝑑𝑡 =�̇�1 𝑋 1𝜃 ∙∙ −�̇�2 𝑋 2𝜃1 ∙
𝑑𝜃12
𝑑𝑡 =�̇�2 𝑋 2𝜃1 ∙
Multiple substrates
26 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
𝜃12
𝜃1 ∙
𝑋 1
𝑋 2
𝑃𝑑𝜃∙ ∙
𝑑𝑡 =− �̇�1 𝑋 1𝜃∙ ∙+�̇� 𝜃12
𝑑𝜃1 ∙
𝑑𝑡 =�̇�1 𝑋 1𝜃 ∙∙ −�̇�2 𝑋 2𝜃1 ∙
𝑑𝜃12
𝑑𝑡 =�̇�2 𝑋 2𝜃1 ∙ −�̇� 𝜃 12
Multiple substrates
27 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
𝜃12
𝜃1 ∙
𝑋 1
𝑋 2
𝑃0=− �̇�1 𝑋 1𝜃∙ ∙+�̇� 𝜃12
0=�̇�1 𝑋 1𝜃∙∙ − �̇�2 𝑋 2𝜃1 ∙
0=𝑋 2𝜃1 ∙− �̇� 𝜃12
Multiple substrates
28 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
𝜃12
𝜃1 ∙
𝑋 1
𝑋 2
𝑃0=− �̇�1 𝑋 1𝜃∙ ∙+�̇� 𝜃12
0=�̇�1 𝑋 1𝜃∙∙ − �̇�2 𝑋 2𝜃1 ∙
1=𝜃 ∙∙+𝜃1 ∙+𝜃12 Sequential complementary
Multiple substrates
29 𝐸 𝑉𝑋
Two substrates:
7th DEB Course, 15-23 April 2013
𝑋 1 �̇�𝐴
𝑋 2
𝜃∙∙
𝜃12
𝜃1 ∙ 𝜃∙2
𝑋 1 𝑋 2
𝑋 2 𝑋 1
𝑃
Parallel complementary
Multiple substrates
30 𝐸 𝑉𝑋
There are several situations that can be seen as a multiple substrate situation:
- Diet - Nutrients- Inhibition-…
Note the limiting cases.
7th DEB Course, 15-23 April 2013
Multiple reserves
31 𝐸 𝑉𝑋
There are cases when the organism accumulates different nutrients and there is a need for multiple reserves.
𝐸𝑉
𝑋 �̇�𝐸 ,𝐴�̇�𝐸 ,𝐷
�̇�𝐸 ,𝐺�̇�𝑉 ,𝐺
7th DEB Course, 15-23 April 2013
Multiple reserves
32 𝐸 𝑉𝑋
𝐸 1𝑉
𝑋 1 �̇�𝐸1 , 𝐴�̇�𝐸1 ,𝐷
�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺
𝑋 2
7th DEB Course, 15-23 April 2013
Multiple reserves
33 𝐸 𝑉𝑋
𝐸 1𝑉
𝑋 1 �̇�𝐸1 , 𝐴�̇�𝐸1 ,𝐷
�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺
𝐸2𝑋 2 �̇�𝐸 2 ,𝐴
7th DEB Course, 15-23 April 2013
Multiple reserves
34 𝐸 𝑉𝑋
The main scheme is done. Let us now look at some important details.
𝐸 1𝑉
𝑋 1 �̇�𝐸1 , 𝐴�̇�𝐸1 ,𝐷
�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺
𝐸2𝑋 2 �̇�𝐸 2 ,𝐴
�̇�𝐸 2 ,𝐷
�̇�𝐸 2 ,𝐺
7th DEB Course, 15-23 April 2013
Multiple reserves
35 𝐸 𝑉𝑋
�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺
�̇�𝐸 2 ,𝐺
7th DEB Course, 15-23 April 2013
Growth SU:
Multiple reserves
36 𝐸 𝑉𝑋
�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺
�̇�𝐸 2 ,𝐺
7th DEB Course, 15-23 April 2013
�̇�𝑉 ,𝐺=(( �̇�𝐸 1 ,𝐺
𝑦𝐸1𝑉)
−1
+( �̇�𝐸 2 ,𝐺
𝑦𝐸2𝑉)
−1
−( �̇� 𝐸1 ,𝐺
𝑦𝐸1𝑉+�̇�𝐸 2 ,𝐺
𝑦𝐸2𝑉)
−1)− 1
Growth SU:
Parallel complementary
Multiple reserves
37 𝐸 𝑉𝑋
Computation of the mobilization fluxes:
�̇�𝑉𝐺�̇�𝐸 ,𝐶 �̇�𝐸𝐺
7th DEB Course, 15-23 April 2013
𝐸𝑉
𝑋 �̇�𝐸 ,𝐴�̇�𝐸 ,𝐷
�̇�𝐸 ,𝐺�̇�𝑉 ,𝐺
Multiple reserves
38 𝐸 𝑉𝑋
Computation of the mobilization fluxes:
�̇�𝑉𝐺�̇�𝐸𝐶 �̇�𝐸𝐺
7th DEB Course, 15-23 April 2013
�̇�𝐸𝐺=𝜅 �̇�𝐸𝐶 −�̇�𝐸𝑆
�̇�𝑉𝐺=�̇�𝐸𝐺 / [𝐸𝐺 ]
�̇�𝐸𝐶=𝐸 (�̇� /𝐿− �̇� )
�̇�=[�̇�𝑉𝐺 ]and
Multiple reserves
39 𝐸 𝑉𝑋
Computation of the mobilization fluxes:
�̇�𝑉𝐺�̇�𝐸𝐶 �̇�𝐸𝐺
7th DEB Course, 15-23 April 2013
�̇�𝐸𝐶=𝐸 [𝐸𝐺 ] �̇� /𝐿+ [�̇�𝐸𝑆 ]𝜅 [𝐸 ]+ [𝐸𝐺 ]
Multiple reserves
40 𝐸 𝑉𝑋
Computation of the mobilization fluxes
𝐸 1𝑉
𝑋 1 �̇�𝐸1 , 𝐴�̇�𝐸1 ,𝐷
�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺
𝐸2𝑋 2 �̇�𝐸 2 ,𝐴
�̇�𝐸 2 ,𝐷
�̇�𝐸 2 ,𝐺
�̇�𝑉 ,𝐺�̇�𝐸𝑖 ,𝐶 �̇�𝐸𝑖 ,𝐺
7th DEB Course, 15-23 April 2013
Multiple reserves
41 𝐸 𝑉𝑋
Computation of the mobilization fluxes
�̇�𝑉 ,𝐺�̇�𝐸𝑖 ,𝐶 �̇�𝐸𝑖 ,𝐺
7th DEB Course, 15-23 April 2013
We can’t break the cycle and compute analytically.
Multiple reserves
42 𝐸 𝑉𝑋
Rejection fluxes
𝐸 1𝑉
𝑋 1 �̇�𝐸1 , 𝐴�̇�𝐸1 ,𝐷
�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺
𝐸2𝑋 2�̇�𝐸 2 ,𝐷
�̇�𝐸 2 ,𝐺
�̇�𝐸1 ,𝑅
7th DEB Course, 15-23 April 2013
�̇�𝐸 2 ,𝐴
Multiple reserves
43 𝐸 𝑉𝑋
Rejection fluxes
𝐸 1𝑉
𝑋 1 �̇�𝐸1 , 𝐴�̇�𝐸1 ,𝐷
�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺
𝐸2𝑋 2 �̇�𝐸 2 ,𝐴
�̇�𝐸 2 ,𝐷
�̇�𝐸 2 ,𝐺
�̇�𝐸1 ,𝑅
7th DEB Course, 15-23 April 2013
Multiple reserves
44 𝐸 𝑉𝑋
Rejection fluxes
𝐸 1𝑉
𝑋 1 �̇�𝐸1 , 𝐴�̇�𝐸1 ,𝐷
�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺
𝐸2𝑋 2 �̇�𝐸 2 ,𝐴
�̇�𝐸 2 ,𝐷
�̇�𝐸 2 ,𝐺
�̇�𝐸1 ,𝑅
7th DEB Course, 15-23 April 2013
Multiple reserves
45 𝐸 𝑉𝑋
Multiple limitation
The case of a 3 reserve (C, N, P) model for microalgae.
Lorena (2008)
N P C
7th DEB Course, 15-23 April 2013
Multiple structures
46 𝐸 𝑉𝑋
In cases where it is important to model an organ that doesn’t grow proportional to the rest of the body, it may be needed to add a new structure.
𝐸 𝑉 1(1 −𝜅) �̇�𝐶
�̇�𝐺1𝜅 �̇�𝐶
�̇�𝑆 1
7th DEB Course, 15-23 April 2013
Multiple structures
47 𝐸 𝑉𝑋
In cases where it is important to model an organ that doesn’t grow proportional to the rest of the body, it may be needed to add a new structure.
𝐸 𝑉 1(1 −𝜅) �̇�𝐶
�̇�𝐺1
𝑉 2
𝜅 �̇�𝐶
�̇�𝑆 1
7th DEB Course, 15-23 April 2013
Multiple structures
48 𝐸 𝑉𝑋
In cases where it is important to model an organ that doesn’t grow proportional to the rest of the body, it may be needed to add a new structure.
𝐸 𝑉 1(1 −𝜅) �̇�𝐶
�̇�𝐺1
𝑉 2
𝜅 𝜅1�̇�𝐶
𝜅(1 −𝜅¿¿1) �̇�𝐶¿
�̇�𝑆 1
7th DEB Course, 15-23 April 2013
Multiple structures
49 𝐸 𝑉𝑋
In cases where it is important to model an organ that doesn’t grow proportional to the rest of the body, it may be needed to add a new structure.
𝐸 𝑉 1(1 −𝜅) �̇�𝐶
�̇�𝐺1
𝑉 2
𝜅 𝜅1�̇�𝐶
𝜅(1 −𝜅¿¿1) �̇�𝐶¿ �̇�𝐺2
�̇�𝑆 1
�̇�𝑆 2
7th DEB Course, 15-23 April 2013
Multiple structures
50 𝐸 𝑉𝑋
Reserves Structures Uptake kappas
Static 1 set Constant + other sGV V GV
7th DEB Course, 15-23 April 2013
Multiple structures
51 𝐸 𝑉𝑋
Reserves Structures Uptake kappas
Static1 set
Constant
Dynamic Dependent onuptake
+ other sGV VGV
other sV
7th DEB Course, 15-23 April 2013
Multiple structures
52 𝐸 𝑉𝑋
Reserves Structures Uptake kappas
Static1 set
Constant
Dynamic Dependent onuptake
Plant (Symbiosis)
m sets Independentuptakes Constant
+ other sGV V
GV
other sV
m sV
7th DEB Course, 15-23 April 2013
Multiple modules
53 𝐸 𝑉𝑋
This case of multiple structures I prefer to call multiple modules.
In this case each module has a set of reserves and one structure and there is translocation between the reserves.(-rule and rejected flux).
7th DEB Course, 15-23 April 2013
Multiple modules
54 Examples of building blocks:
7th DEB Course, 15-23 April 2013
Multiple modules
55 𝐸 𝑉𝑋
7th DEB Course, 15-23 April 2013
Multiple maturities
56 𝐸 𝑉𝑋
Regrowth of a chestnut tree
Does it make sense to talk about multiple maturities?
I’m not aware of any simulation made with multiple maturities, but it could make sense.
7th DEB Course, 15-23 April 2013
Multiple products
57 𝐸 𝑉𝑋
Does it make sense to use more fluxes for product formation?
7th DEB Course, 15-23 April 2013
𝐸 𝑉𝑋 �̇�𝐴�̇�𝐷 �̇�𝐺
Life Engine
58What is an engine?A toolbox used in the making of a videogame, that allows for the improvement of realism in videogames
Physics Behaviour
7th DEB Course, 15-23 April 2013
Life Engine
59What is an engine?A toolbox used in the making of a videogame, that allows for the improvement of realism in videogames
Life EngineVideogame engine based on DEB which aims to improve the realismof the biological/metabolic features of a game and increase the automatic generation of novelty
DEBlibC++ library of DEB that will be used by Life Engine
Scientific interfaceWe also want to have a user-friendly scientific interface that uses DEBlib
7th DEB Course, 15-23 April 2013
Generalized DEB model
60
To simulate a given organism we have to define• Number of modules and type of translocation
(Plant has two modules – Root and Shoot)• Number of state variables by type in a module
(which fixes the number of parameters by type)• Values of the parameters• Initial values of the state variables
... and we’re set to go.
7th DEB Course, 15-23 April 2013
DEBlib architecture
61 Flexibility in organism definition allows for a balance between speed and the needs of the user:Single-part organism• Standard organism: the number of reserves, structures and products is fixed at
(1, 1, 1) in compile-time.• Complex organism: the number of reserves, structures and products is is
different of (1, 1, 1) and it is defined and fixed in compile-time.• Dynamic (complex) organism: the number of reserves, structures and
products is can be (1, 1, 1) or different and is defined only in run-time.Multipart organism• Multipart organism: the organism is defined as a set of sub-organisms of the
type single-part with energy and mass translocations; the number and type of sub-organisms is defined and fixed in compile-time.
• Dynamic multipart organism: the organism is defined as a set of sub-organisms of the type single-part with energy and mass translocations; the number and type of sub-organisms is defined only in run-time.
7th DEB Course, 15-23 April 2013
DEBlib architecture
62
DEB model
Food web
Organism
Ecosystem
7th DEB Course, 15-23 April 2013
DEBlib architecture
63 Policies
Morphism: isomorph, V1-morph, V0-morph (no change between morphisms is available)
Assimilation: functional response, handling
Ageing: DEB (with acceleration, hazard and survival probability) or life expectancy
Reproduction: ability, gestation
Life cycle: it is possible to define life stages and transitions (with impacts in metabolism)
7th DEB Course, 15-23 April 2013
Use cases
64 Dictyostelium discoideum (work with M. Rodrigues)
• 2D environment• Movement
dependent on food
7th DEB Course, 15-23 April 2013
Use cases
65 Dictyostelium discoideum (work with M. Rodrigues)
• 2D environment• Movement
dependent on food • Aggregation
behaviour triggered by starvation
7th DEB Course, 15-23 April 2013
Use cases
66 Escherichia coli (work with S. Cruz)
• ION framework• 3D environment• Movement
dependent on food: probability of tumble and run function of food gradient
7th DEB Course, 15-23 April 2013
Use cases
67 Escherichia coli (work with S. Cruz)
• ION framework• 3D environment• Movement
dependent on food • Substrate presented
in blue
7th DEB Course, 15-23 April 2013
Use cases
68 Escherichia coli (work with S. Cruz)
• ION framework• 3D environment• Movement
dependent on food • Substrate presented
in blue
7th DEB Course, 15-23 April 2013
Use cases
69 Escherichia coli (work with S. Cruz)
7th DEB Course, 15-23 April 2013
Use cases
70 Escherichia coli (work with S. Cruz)
7th DEB Course, 15-23 April 2013
Multiple products
71 𝐸 𝑉𝑋
Microalgae models
Life stagesTransformations and products
7th DEB Course, 15-23 April 2013
𝐸 𝑉𝑋 �̇�𝐴�̇�𝐷 �̇�𝐺
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