Lecturer: Bas Kooijman
Affiliation: Dept Theoretical Biology, Vrije Universiteit, Amsterdam
3rd AQUAdeb meeting, Rennes
Date: 2007/12/18
Title: What the egg can tell about its hen

Abstract: The parameter estimation exercise during the 2nd AQUAdeb
meeting in Nantes revealed some problems which turned out to have two causes
1) particular combination of easy-to-measure quantities are impossible 
   in DEB theory
2) the calculation of the initial amount of reserve (of an egg) turned
   out to be unstable; a shooting method was used for a set of 3 ode's
   (reserve, structure, maturity)
   (ode = ordinary differential equation)

Meanwhile I solved these problems, i.e.
1) impossible combinations of values frequently had too large values for
   the age at birth. I now have derived a maximum value for this quantity,
   which turned out the ratio of the length at birth as fraction of the 
   ultimate length and the von Bertalanffy growth rate.
2) the (explicit) expressions for the initial amount of reserve and the age 
   at birth as reported in the book turned out to apply even when the 
   maintenance ratio (i.e. the ratio of the maturity and somatic 
   maintenance rate coefficients) is not equal to one. The problem is in the
   length at birth. I found a (rather strange) scaling which reduced the 
   set of 3 ode's to a single one that can be solved analytically. I wrote 
   three routines in DEBtool to solve the length at birth (varying from fast 
   and dirty to slow and clean). These routines are both much faster and much 
   more stable than the original shooting method with 3 ode's

I rewrote the get_pars routines for DEBtool/animal, which are now
faster and more robust, and extended the set of get_pars routines to
cover all combinations of single and multiple food levels, growth and
growth + reproduction data. I also rewrote the routines in DEBtool/tox
for effects on reproduction. The earlier version considered the
maximum reproduction rate as a parameter that can be affected by
toxicants, for simplicity's sake. The new one is better consistent
with core DEB theory, but is also more computationally demanding. We
needed this extension for our work on effects of mixtures (the
Nomiracle EU-project). Another example where simplification gave problems.

For simplicity's sake, the deb-book focuses on the special case that
the maintenance ratio equals one, which makes that stage transitions
occur at a body size that is independent of food availability. I now
studied some implications of the extention to other values for the
maintenance ratio, when the size at stage transition does depend on
food availability. Some results are
1) when toxic compounds increase the energy cost for structure, the
   allocation to reproduction as well as the size at birth decreases
   such that the reproduction rate (eggs per time) can increase for
   low concentrations (hormesis effect). Other modes of action don't
   have this effect. The cause of hormesis is typically enigmatic, but
   this might be one of the mechanisms that needs to be verified against
   data.
2) when cells in the early embryonic stages detach, 2, 4 and sometimes 
   even 8 neonates can be born, typically of reduced size. We can now
   evaluate how much the size will be reduced. It turns out that this type
   of twinning can only occur in species in which the (ultimate) body size 
   is large enough. Daphnia magna is too small for this; given its 
   parameters it can only occur in species with 1.7 times maximum body 
   length of D. magna. A nice prediction that needs verification against data.
3) the size at stage transition depends sensitively on the maintenance ratio.
   This life-history characteristic varies a lot among taxa, and typically 
   deviates much more for body size scaling predictions than physiological
   parameters (such the respiration rate or the von Bertalanffy growth rate).
   One of the reasons of these deviations might be variation of the value of 
   the maintenance ratio. A nice new insight, with which I am very happy.

More information about research on DEB theory can be found at
  http://www.bio.vu.nl/thb/deb/

An ecological introductory paper to the DEB theory can be found at
  http://www.bio.vu.nl/thb/research/bib/NisbMull2000.html
Aa biological introductory paper to the DEB theory can be found at
  http://www.bio.vu.nl/thb/research/bib/Kooy2001.html
The standard model and parameter estimation is discussed in
  http://www.bio.vu.nl/thb/deb/essays/Meer2006.html
A formal presentation is given in  
  http://www.bio.vu.nl/thb/research/bib/SousDomi2008.html