multivariate deb models generalization of the standard deb model with multiple state variables

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,[email protected]

7th DEB Course

TEXEL, 15-23 April

mailto:[email protected]

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


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.


State variables



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.


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.


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.


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.


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.


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


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.


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


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


Multiple substrates

16 𝐸 𝑉𝑋

𝑋

One substrate:

𝑋 b

𝑋 p

𝑋 �̇�𝐴�̇�𝑋


�̇�𝑋

�̇�𝐴

𝑋

Multiple substrates

17 𝐸 𝑉𝑋

𝑋

One substrate:


�̇�𝐴 𝑋→𝑃

Multiple substrates

18 𝐸 𝑉𝑋

Two substrates:


𝑋 1 �̇�𝐴

𝑋 2

𝑋 1+ 𝑋 2→𝑃

complementary

Multiple substrates

19 𝐸 𝑉𝑋

Two substrates:


𝑋 1 �̇�𝐴

𝑋 2

𝜃∙∙

Multiple substrates

20 𝐸 𝑉𝑋

Two substrates:


𝑋 1 �̇�𝐴

𝑋 2

𝜃∙∙

𝜃1 ∙

𝑋 1

Multiple substrates

21 𝐸 𝑉𝑋

Two substrates:


𝑋 1 �̇�𝐴

𝑋 2

𝜃∙∙

𝜃12

𝜃1 ∙

𝑋 1

𝑋 2

Multiple substrates

22 𝐸 𝑉𝑋

Two substrates:


𝑋 1 �̇�𝐴

𝑋 2

𝜃∙∙

𝜃12

𝜃1 ∙

𝑋 1

𝑋 2

𝑃

Multiple substrates

23 𝐸 𝑉𝑋

Two substrates:


𝑋 1 �̇�𝐴

𝑋 2

𝜃∙∙

𝑑𝜃∙ ∙

𝑑𝑡 =¿

𝑑𝜃1 ∙

𝑑𝑡 =¿

𝑑𝜃12

𝑑𝑡 =¿

𝜃12

𝜃1 ∙

Multiple substrates

24 𝐸 𝑉𝑋

Two substrates:


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:


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:


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:


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:


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:


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.


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


Multiple reserves

32 𝐸 𝑉𝑋

𝐸 1𝑉

𝑋 1 �̇�𝐸1 , 𝐴�̇�𝐸1 ,𝐷

�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺

𝑋 2


Multiple reserves

33 𝐸 𝑉𝑋

𝐸 1𝑉

𝑋 1 �̇�𝐸1 , 𝐴�̇�𝐸1 ,𝐷

�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺

𝐸2𝑋 2 �̇�𝐸 2 ,𝐴


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


Multiple reserves

35 𝐸 𝑉𝑋

�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺

�̇�𝐸 2 ,𝐺


Growth SU:

Multiple reserves

36 𝐸 𝑉𝑋

�̇�𝐸1 ,𝐺�̇�𝑉 ,𝐺

�̇�𝐸 2 ,𝐺


�̇�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


𝐸𝑉

𝑋 �̇�𝐸 ,𝐴�̇�𝐸 ,𝐷

�̇�𝐸 ,𝐺�̇�𝑉 ,𝐺

Multiple reserves

38 𝐸 𝑉𝑋


�̇�𝑉𝐺�̇�𝐸𝐶 �̇�𝐸𝐺


�̇�𝐸𝐺=𝜅 �̇�𝐸𝐶 −�̇�𝐸𝑆

�̇�𝑉𝐺=�̇�𝐸𝐺 / [𝐸𝐺 ]

�̇�𝐸𝐶=𝐸 (�̇� /𝐿− �̇� )

�̇�=[�̇�𝑉𝐺 ]and

Multiple reserves

39 𝐸 𝑉𝑋


�̇�𝑉𝐺�̇�𝐸𝐶 �̇�𝐸𝐺


�̇�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


Multiple reserves

41 𝐸 𝑉𝑋

Computation of the mobilization fluxes

�̇�𝑉 ,𝐺�̇�𝐸𝑖 ,𝐶 �̇�𝐸𝑖 ,𝐺


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


�̇�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


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


Multiple reserves

45 𝐸 𝑉𝑋

Multiple limitation

The case of a 3 reserve (C, N, P) model for microalgae.

Lorena (2008)

N P C


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


Multiple structures

47 𝐸 𝑉𝑋


𝐸 𝑉 1(1 −𝜅) �̇�𝐶

�̇�𝐺1

𝑉 2

𝜅 �̇�𝐶

�̇�𝑆 1


Multiple structures

48 𝐸 𝑉𝑋


𝐸 𝑉 1(1 −𝜅) �̇�𝐶

�̇�𝐺1

𝑉 2

𝜅 𝜅1�̇�𝐶

𝜅(1 −𝜅¿¿1) �̇�𝐶¿

�̇�𝑆 1


Multiple structures

49 𝐸 𝑉𝑋


𝐸 𝑉 1(1 −𝜅) �̇�𝐶

�̇�𝐺1

𝑉 2

𝜅 𝜅1�̇�𝐶

𝜅(1 −𝜅¿¿1) �̇�𝐶¿ �̇�𝐺2

�̇�𝑆 1

�̇�𝑆 2


Multiple structures

50 𝐸 𝑉𝑋

Reserves Structures Uptake kappas

Static 1 set Constant + other sGV V GV


Multiple structures

51 𝐸 𝑉𝑋


Static1 set

Constant

Dynamic Dependent onuptake

+ other sGV VGV

other sV


Multiple structures

52 𝐸 𝑉𝑋


Static1 set

Constant

Dynamic Dependent onuptake

Plant (Symbiosis)

m sets Independentuptakes Constant

+ other sGV V

GV

other sV

m sV


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).


Multiple modules

54 Examples of building blocks:


Multiple modules

55 𝐸 𝑉𝑋


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.


Multiple products

57 𝐸 𝑉𝑋

Does it make sense to use more fluxes for product formation?


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


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


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.


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.


DEBlib architecture

62

DEB model

Food web

Organism

Ecosystem


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)


Use cases

64 Dictyostelium discoideum (work with M. Rodrigues)

• 2D environment• Movement

dependent on food


Use cases

65 Dictyostelium discoideum (work with M. Rodrigues)

• 2D environment• Movement

dependent on food • Aggregation

behaviour triggered by starvation


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


Use cases



dependent on food • Substrate presented

in blue


Use cases



Multiple products

71 𝐸 𝑉𝑋

Microalgae models

Life stagesTransformations and products


𝐸 𝑉𝑋 �̇�𝐴�̇�𝐷 �̇�𝐺

multivariate deb models generalization of the standard deb model with multiple state variables

Documents

reserve mobilization

deb course texel

organismreserve assimilation

deb theory

metabolic purpose

generalized deb models

state variables2

multiple maturities