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Reduction of acetic acid production during wine fermentation by Saccharomyces cerevisiae x Saccharomyces kudriavzevii hybrids using adaptive evolution under lipids limitation

Amandine Deroite, Jean-Luc Legras, Anne Ortiz-Julien, Sylvie Dequin

To cite this version:

Amandine Deroite, Jean-Luc Legras, Anne Ortiz-Julien, Sylvie Dequin. Reduction of acetic acid production during wine fermentation by Saccharomyces cerevisiae x Saccharomyces kudriavzevii hybrids using adaptive evolution under lipids limitation. ISSY 34, Oct 2018, Bariloche, Argentina. hal-01982725

REDUCTION OF ACETIC ACID PRODUCTION DURING WINE FERMENTATION BY SACCHAROMYCES CEREVISIAE X SACCHAROMYCES KUDRIAVZEVII HYBRIDS USING ADAPTIVE EVOLUTION UNDER LIPIDS LIMITATION

Amandine Deroite 1,2 , Jean-Luc Legras 2 , Anne Ortiz-Julien 1 , Sylvie Dequin 2 1 Lallemand SAS, F-31700 Blagnac, France, 2 SPO, Univ Montpellier, INRA, Montpellier SupAgro, Montpellier.

Presenting author: deroitea@supagro.inra.fr S. cerevisiae x S. kudriavzevii hybrids are typically used for white wine fermentation because of their cryotolerance. One group of these hybrids presenting a unique ability to release thiols varietal aromas produces as well unacceptable amounts of acetic acid under specific conditions, which is detrimental for wine organoleptic quality. The objective of this work is to reduce this acetic acid production through an adaptive evolution strategy. A first comparison of the different strains for their production of acetic acid revealed the presence of two groups of strains, further called High (HAP) and Low (LAP) acetic acid producers. When comparing the genomes of the different strains, two genetic groups corresponding to the level of acetic acid production were revealed. HAP strains have lost copies of the region C, while LAP present a different balance of chromosome copy number of each species and both of these modifications may contribute to differences in acetic acid production. Thanks to a Box Behnken experimental design aiming to study the impact of environmental conditions on acetic acid production, we showed that lipids modulate acetic acid and thiols production for both LAP and HAP S. cerevisiae X S. kudriavzevii hybrids during wine fermentation. Based on this conclusion, we used an adaptive evolution strategy based on a long-term batch culture under lipids and oxygen limitation in order to reduce the acetic acid production of those hybrids.

After 250 generations, we selected several evolved strains able to release lower amounts of acetic acid in wine while keeping high thiols liberation ability. The genome of these evolved strains will be sequenced in order to identify the mutations involved in the phenotype.


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1

PDG Documentation, version 2

Sylvie Muratorio, July 2017 1) PDG general purpose and main originalities 2.1.

General purpose The Physio-Demo-Genetics model (denoted PDG) is an individual-based, spatially explicit mechanistic simulation model, coupling (1) a physiological module simulating individual tree responses to the environment (the CASTANEA model, Dufrêne et al. 2005); (2) a demographic module simulating tree survival, reproduction and pollen and seed dispersal; and (3) a quantitative genetics module controlling the heritability of key life history traits. The initial objectives of PDG were twofold: first, PDG aims at simulating the temporal and spatial variation of life-history traits in a tree population within a heterogeneous environment, while accounting for genetic and plastic components of this variation; we focus here on those functional traits that mediate individual tree response to climate, and whose values are computed by the CASTANEA model. Secondly, PDG allows to investigate how this interindividual variation affect adaptive tree population dynamics across generations. PDG model runs on a yearly time step, although its physiological module (CASTANEA) runs on a daily time step. PDG model was used to investigate the plastic and genetic components of the variations of the timing of budburst along an altitudinal gradient of Fagus sylvatica (the European beech), in the reference publication of PDG (Oddou-Muratorio & Davi 2014).

PDG takes advantage of the modularity of the Capsis platform. The physiological module is based on the CASTANEA library, implemented by Hendrik Davi in Capsis. The quantitative genetics module is based on the GENETICS library implemented by Ingrid Seynave and Christian Pichot in Capsis (Seinave & Pichot 2004). The framework of the demographic module is similar to the one used in other previous models of Capsis (eg Luberon, Alisier).

This documentation first details the main originalities of PDG, that is the modeling of plastic and genetic variation in functional traits (paragraph 1.2 below) and the mortality process (paragraph 1.3 below). In paragraph 2, the general principle of use of PDG is described, while paragraph 3 describe the architecture and implementation of PDG.

2.2.

Modeling plastic and genetic variation of life-history traits Presently, PDG allows two life-history traits to be considered as both plastically and genetically variable: the timing of budburst (TBB, as in the reference publication) and water use efficiency (WUE). Potentially, any other trait modeled in CASTANEA can also be considered, but this will require informatics development by the user. PDG uses a classical quantitative genetic framework to model phenotypic and genetic variation for TBB and WUE.

Model for TBB:

In CASTANEA, the timing of budburst (TBB) is usually simulated following (eq. 9-11 in Dufrêne et al. 2005):

           1 T T if 0 1 T T if T - T R 2 2 2 frcBB Nstart N or Nstart N and
(Eq 1)

where RfrcBB is the rate of forcing for bud break, T the mean daily temperature, T2 the base temperature, N the day of year and NSTART1 the date of onset of rest. if S frcBB ³ F critBB (Eq 3) with SfrcBB the state of forcing, FcritBB the critical value of state of forcing for the transition from quiescence to the active period and TBB the day when bud break occurred. Note that T2, Nstart1 and FCRITBB are parameters, while RfrcBB, SfrcBB and TBB are dynamic variables of CASTANEA.

The simple one-phase phenological sub-model described by the equations 1-3 assumes that the variation is TBB is driven only by forcing temperatures; this hypothesis is valid for our case study (Fagus sylvatica on Mont Ventoux), but should be considered with caution in other cases. Note that diverse refined phenology models (including chilling) were recently integrated from Phenofit into CASTANEA, and could allow different hypothesis to be made.

The main originality of PDG is to consider FCRITBB parameter as genetically determined and variable among individuals. The value of FcritBB is determined by ten independent biallelic loci with purely additive effects (see Appendix 1). The inter-individual variation in TBB thereby results from the inter-individual variation of mean daily temperature (T in equation 1) and from genotypic variation in FcritBB. This is the classical frame of quantitative genetics; note that we did not explicitly introduce genotype-by-environment interaction.

Figure 1A illustrates the variation of TBB along a temperature gradient as simulated in PDG. A given genotype (one color curve on Figure 1A) unfold its leaves earlier when environment get hotter because the sum of temperature it receives (i.e. the state of forcing SfrcBB ) reaches faster FCRITBB. On the other hand, in a given environment, late and early individuals can be observed as the result of genetic variation for FCRITBB: late individuals typically require higher FCRITBB than early individuals (two different color curves on Figure 1A).

Model for WUE: water use efficiency (WUE) can be defined at leaf level as the ratio of assimilation (A) to stomatal conductance (gsH2O):

𝑊𝑈𝐸 𝑖 = 𝐴/𝑔 𝑠𝐻 2 𝑂 (Eq 4)
In CASTANEA, three main equations determine assimilation A and gsH2O (eq 1,2 and 4 in Dufrêne et al. 2005), following the model from Ball et al. (1987). First the carbon dioxide demand is: 𝐴 = 𝑉 𝐶 -𝑅 𝑑 . (Eq 5)

where VC is the carboxylation rate and Rd the respiration during the night. Second, the carbon dioxide supply writes:

𝐴 = 𝑔 𝑠𝐶𝑂 2 (𝐶 𝑆 -𝐶 𝐼 ) . . (Eq 6)
where gsCO2 is the stomatal conductance for CO2 and (CS-CI) the gradient of CO2 between evaporative site and leaf surface.

Third, the carbon dioxide control which allows the calculation of gsH2O writes:

𝑔 𝑠𝐻 2 𝑂 = 𝑔 0 +𝑔 1 ×𝐴×𝑅𝐻 𝐶 𝑆 (Eq 7)
where is RH the relative humidity in the surrounding air, g1 depends on soil water stress and g0 correspond to cuticle conductance of the leaf.

Re-arranging eq 7 shows that g1, the slope of the relation between photosynthesis and stomatal conductance, is inversely related to WUE :

𝐴 𝑔 𝑠𝐻 2 𝑂- 𝑔 0 𝐶 𝑆 ⁄ = 𝐶 𝑆 𝑔 1 ×𝑅𝐻 .. ( Eq
8)

Assuming

, the higher g1, the lower WUE.

In CASTANEA, g1 is assumed to decrease linearly when soil water storage decreases. The effect of soil water stress on photosynthesis is mediated through g1:

𝑔 1 = (𝑔 1𝑚𝑎𝑥 -𝑔 1𝑚𝑖𝑛 ) × 𝑟𝑒𝑑𝑢𝑐 + 𝑔 1𝑚𝑖𝑛 . (Eq 9)
Where g1max and g1min are the maximal and minimal values taken by g1, and "reduc" is a soil water stress index varying between 0 (maximal soil water stress) and 1 (no water stress).

In PDG, we consider g1max as a parameter genetically determined and variable among individuals. The value of g1max is determined by ten independent biallelic loci with purely additive effects (see Appendix 1). The inter-individual variation in WUE thereby results from the inter-individual variation of soil water stress (reduc in eq 9) and from genotypic variation in g1max. This is again the classical frame of quantitative genetics; note that as for TBB, we did not explicitly introduce genotype-by-environment interaction.

Figure 1B illustrates the variation of WUE along a water tress gradient as simulated in PDG. A given genotype (one color curve on Figure 1B) favors water conservation when water stress increases because g1 decreases as "reduc" increases. On another hand, in a given environment, individuals favoring either water uptake or water conservation can be observed as the result of genetic variation for g1max: individuals with high g1max typically favor water uptake while individuals with low g1max typically favor water conservation (two different color curves on Figure 1B).

2.3.

Modeling mortality The reserves produced by photosynthesis at a daily time step are allocated to growth and the level of reserve together with tree water potential are used to compute the probability of an adult tree to die. There are three different thresholds associated with mortality:

 Below a critical value of carbon reserve at the end of the year, there is a non-zero probability for the tree to die because of carbon starvation (PmortCS)  Below a critical value of carbon reserve at budflush, there is a non-zero probability for the tree to die because of carbon starvation (PmortBF)

 Below a critical daily value of water potential there is a non-zero probability for the tree to die because of hydraulic failure (PmortHF) Moreover, mortality is a stochastic process.

2) General principle of use 2.1.

Initialization of the simulations To run PDG, the user must provide:

 A species parameter file specifying SOME Castanea parameters, ie all the parameters defined at tree species level. This file is named CastaneaSpecies3.txt, its values should not be changed and it is located in the directory data/physiodemogenetics/  An inventory file containing:  parameters related to the simulation design  parameters of the demographic module  parameters of the genetic module, in particular the genetic and environmental controls of FCRITBB and g1max (heritability, allelic effects), as well as on the genetic map of the considered species.  parameters of CASTANEA which are not defined in the CastaneaSpecies3.txt file, ie all the parameters related to simulation option in CASTANEA, as well as parameters related to plot (soil etc..)  definition of the plot (list of cells with their properties),  a list of initial trees with their properties at the beginning of the simulation (year y0);

Note that all the trees are individual trees (no multipleGenotype allowed); depending on their age, trees are considered either as adult or seedlings (by default, the age at which tree become adult, adultAge, is 40 years old). In the present version of PDG, all the trees belong to the same single species (see Appendix 2 for the list of possible species to consider).

The inventory file should be located in the directory data/physiodemogenetics.

 A climate file containing the values of six climatic variables (global radiation, rainfall, wind speed, air humidity and temperature) on a daily basis. This file should be located in the directory data/physiodemogenetics/climate. All these files, as well as the list of parameters to control are detailed in Appendix 2.

2.2.

Running a simulation Originally, PDG was developed to run simulation in even-aged populations, containing either only "active" adult trees, or "active" seedlings ("active" meaning growing and eventually reproducing trees). For such even-aged simulations, the user can run "evolution" from an initial year y0 to a next year (yn) using 3 different modes:  mode "g" (for growth): all the trees are considered as adult, and growth and mortality are computed for all the trees of the stand using CASTANEA. Note that neither growth nor mortality are affected by competition (each tree grow as a dominant tree). Although adult trees can produce seeds, there is no recruitment of young seedlings in evolution with mode "g" step.  mode "r" (for reproduction): all the trees are considered as adult and reproductive; growth and mortality are computed for all the trees of the stand using CASTANEA; Seeds are dispersed and eventually germinate, and the newly established young seedlings do not grow/die.  mode "s" (for seedlings): all the trees are considered as seedlings, and growth is simulated only for a subset of them using CASTANEA. Here, we assume that CASTANEA model is valid for seedlings, while it was calibrated for adult trees. The only difference between mode "g" and "s" (besides the subsampling of growing individual) is that seedlings cannot die (the critical value of carbon reserve and of water potential are set to non-limiting values). There is no recruitment of young seedlings during a "s" step (no adult trees present). At the end of phase "s", the dimensions of all the seedlings that did not grow are updated based on the dimension of the seedlings that grew. Then, seedlings densitydependent mortality is simulated, considering either (1) neutral (genotype independent) mortality or (2) genotype-driven mortality. The maximum final density of the stand after a "s" step is D trees/ha (where the parameter D is typically the density of adult trees)

For each evolution step in mode g, r or s, the user has to indicate which climate file is used, and specify the initial year, month and day from which evolution begins.

These different modes of evolution can be combined. For instance, using an inventory file where all the initial trees have an age of 10 years old (seedlings), and the evolution sequence "s30 g50 r60" leads to the following evolution:

 From year y0=0 to year yn=30, a subset of the trees from the initial stand are grown as seedlings (ie without mortality);  At year 30, the dimensions of all seedlings are updated as function of the dimensions of the seedlings that grew, and mortality of seedlings can occur. As the seedlings that survived are 40 years old, they switch to the category "adult trees".  From year 30 to year 50, all of the adult trees grow/die up (reaching age 60).  From year 50 to year 60, all of the adult trees grow/die/reproduce (reaching age 70), and a cohort of new seedlings of age 10 at year 60 will eventually be recruited in the stand. Important note: it is the responsibility of the user to take care that the values of trees characteristics given in the inventory file (for instance their age), the parameters values (for instance adultAge) and the sequence of evolution are consistent one which each other.

There is a fourth possible mode, mode "i" (for intervention), which consist in cutting all the adult trees (with age >40) at a given year yn. Mode "i" thus corresponds to an instant operation, and not to an evolution (no climate required) By combining these four modes, the user can simulate evolution across successive and non-overlapping generations. Table 1 below summarizes a typical sequence of evolution in PDG, which allows to simulate 5 nonoverlapping generations in an even-aged context.

Table 1: sequence of evolution simulated when using the instruction "g20 r30 i30 s60 g80 r90 i90 s120 g140 r150 i150 s180 g200 r210 i210 s240 g260 r270 i270 s300 g320 r330 2: Scheme of the evolution simulated when using the instruction "g20 r30 i30 s60 g80 r90 i90 s120 g140 r150 i150 s180 g200 r210 i210 s240 g260 r270 i270 s300 g320 r330" Another version of PDG for uneven-aged stand is under development, and involves the coupling with the SamsaraLight library.

2.3.

Simulation options PDG can be run either in graphical mode (typically for test or development), or in script mode (advisable for routine simulations). Command line for script mode in Linux:

Pre-evolution : it is theoretically possible to run a neutral pre-evolution, with the objective to bring the initial population to a migration-drift equilibrium. To that aim, the parameter preEvolutionTime should be set to a non-zero integer value (for instance 5) specifying the number of generations of pre-evolution to be run. Preevolution includes only mating (no growth nor mortality) across non-overlapping generations. It was implemented in the context of the study of Oddou-Muratorio and Davi 2014, and has several constraints (see Appendix 2). Presently, we rather advice to use initial stand already at migration-drift equilibrium, or possibly at selection-migration-drift equilibrium using other simulation software to generate inventory files (see Appendix 2).

2.4.

Results of simulations If the inventory file is located there: data/physiodemogenetics/inventory.inv, the main results are stored in the directory data/physiodemogenetics/output-inventory, as several text files:

 The main result file is called yearlyResult (csv file), and can be loaded in R. It contains for each tree and year the output of the CASTANEA simulations  Inventory files can be exported under the name GXAdult.inv (typically t the beginning of each generation X)  The realized fitness of each adult tree of generation X are computed at the number of offspring they have at the beginning of generation X+1; these values are stored in the file YearlyIndivFitness  Several files are exported (typically at the beginning of "r" step) contain the adult tree genotypes and position to be analysed with classical population genetic software (Genepop, Spagedi). All these files are described in Appendix 3.

2.5.

Main limitations The present version of PDG has several limitations:  Each tree bears its own CASTANEA model, which simulates carbon and water fluxes independently from the other trees of the stands; so each tree behaves as a dominant tree.  There is neither mortality due to competition, nor mortality due to perturbation such as storms.  A single tree species can be simulated (no mixture of different species)  Simulations are aimed to follow an "even-aged trajectory", such as can be achieved through sylviculture; but the whole simulated stand correspond to a single management unit and there is no gene flow from other stands.  Typically, to run 5 generations in a stand of 2500 individuals, the simulation time is 2.5 days. The data classes are linked one to each other, and also with classes of CAPSIS belonging to the kernel or to libraries; in particular, the class PDGTree is central in the philosophy of PDG, as it is connected both to the GENETICS and the CASTANEA libraries (Figure 3). Similarly, the process classes of PDG are linked one to each other, as illustrated in Figure 4. 2 and other classes from the GENETICS library (in orange) or from the CASTANEA library (in green), of from CAPSIS kernel (in black)

3) Model architecture and main features

Figure 4: Relationship between PDG process classes listed in Table 2 and other classes from the GENETICS library (in orange) or from the CASTANEA library (in green).

A main originality of PDG model is that there the historical sequences of steps is not stored during and evolution. This means that at a year yn, the user cannot access to previous stands at year <n. All the information has to be exported in files along the simulations.

4) References

Ball In PDG, individual genotype at given set of loci determines the value of two CASTANEA parameters (FCRITBB and g1max) which themselves determine, in interaction with environment, the individual values of the two traits of interest (respectively TBB, the timing of budburst, and WUE, Water Use Efficiency). The inventory file contains all the information related to the relation between genotype and the value of FCRITBB and g1max, as well as the distribution of genotype at initiation (see Appendix 2 for an example of such an inventory file). The objective of this appendix is to 1) introduce notations of for the main quantitative genetic parameters of interest in PDG, 2) relate these parameters to the variables of CAPSIS and 3) describe in practice how FCRITBB and g1max are computed, and related to the list of genotypes given in the inventory file.

1) The quantitative genetic framework 1.1.

From genotype to phenotype: We consider here FCRITBB and g1max as two "parametric traits" to model as function of genotype. Note that FCRITBB and g1max are in fact CASTANEA parameters controlling traits (TBB and WUE, see details below). We assume pure additivity, without dominance nor epistasis. Moreover, we consider that the loci coding for the parametric traits are all bi-allelic (SNP like). In the following, we develop general notations for the relationship between genotype and phenotype at a parametric traits TP. (2001) assume that all ml are equal, and so do we)

Assuming Hardy-Weinberg, the mean genetic value for locus l is : Al = pl ² ( ml + l) + ql ² ( ml -l) +2 plqlml Al = ml (pl ² + ql ² +2 plql ) + (pl ² -ql ² )l Al = ml + (pl ² -ql ² )l = ml + (pl+ ql) (pl-ql)l =ml + (pl-ql)l

Mean genetic value for n loci : A = n Gl = n ml + n (pl -ql) l, with ml=µT/10 (eq A1.1)

Allelic effects, genetic variance and heritability Table A1.2 Definition of allelic effects at locus l

Gamete Genotypes contributed

Mean offspring value Mean effect (deviation from the mean of genotypes) A1 pA1A1 + q A1A2 p(ml + l)+qml = ml +pl A1 = ml +pl -ml -(pl-ql)l= qll A2 pA1A2 + q A2A2 pml+q(ml -l) = ml -ql

A2 = ml -ql-(pl-ql)l= -pll
The parameter l is the mean allelic substitution effect: A1-A2 =l NB Bost shows that there is a constraint on allelic substitution effect: |l | < ml NB allelic effects depend of the allelic frequencies.

Table A1.3 Definition of Breeding values at locus l

Genotype Frequency Mean effect of the alleles of one parent A1A1 p² 2A1 = 2qll A1A2 2pq A1+A2 = l(ql -pl) A2A2 q² 2A2 = -2pll Moyenne pl²(2qll)+2plqll(ql -pl)-2ql²pll = 0 Additive variance VAl 4pl²ql²l² + 2pq(q-p)²l² + 4ql²pl²l² =4p²q²²+2pq 3 ²+2p 3 q²-4 p²q²²+ 4q²p²² =2plqll² Additive variance = variance of breeding values (variance of a centered-scaled variable = expectation of its squared value) For n independent loci, the total additive variance is given as the sum of the contributions at each locus: VA = VAl (eq A1.2)

NB Additive variance is scaled by the allelic effects l²= VAl/2plql

Total phenotypic variation and heritability

The total phenotypic variation for each parametric trait is classically defined as: VP=VA+VD+VE (eq A1.3) where VD is the dominance variance, and VE the environmental variance. In Capsis, we assume that VD =0 The heritability for each parametric trait is classically defined as : h²= VA/VP=1 (eq A1.4)

2) Relation with the variables in Capsis GENETICS library and in PDG model NB: in the following, the names of CAPSIS-GENETICS variables are italicized.

Individual genotypes:

In PDG, each individual tree has got its own genotype, which is an IndividualGenotype. An IndividualGenotype is composed of 3 tables (Figure A1.1).  The first table, named nuclearDNA, is two-dimensional and contains the genotype carried by nuclear DNA. This table consists of n lines (n corresponding to the number of loci) and 2 columns, the first for the mother's DNA and the second for the father's derived DNA. Each cell in the table contains an integer encoding the allele of the locus considered. In PDG, n=2 for all the loci coding for parametric traits.  The second and third arrays (named mCytoplasmicDNA and pCytoplasmicDNA respectively) are unidimensional and contain respectively (i) the genotype carried by cytoplasmic DNA transmitted by the mother, which is contained, depending on the species, in the mitochondria and / or in the chloroplast, and (ii) the genotype carried by cytoplasmic DNA transmitted by the father (mitochondrial and / or chloroplast depending on the species). These tables consist of n lines (n being the number of loci) and a column, each cell containing an integer encoding the allele at that location. Accessors : getGenotype ()

Figure A1.1: Schematic representation of an Individual Genotype

For trees whose species is genetically considered, two of the three tables may be empty. All trees of the same species must have the same genotype format, ie (i) the number of loci studied by DNA type (nuclear, maternal and paternal cytoplasmic) is identical for all trees of the same species and (ii) the loci are always in the same order.

For the trees of the initial stand, these genotypes have to be provided in the inventory file (see example in appendix 2). For trees generated during simulations in PDG, the genotype is computed by the GENETICS library, through meiosis and mating models.

Description of the population-level genetic composition

CAPSIS genetic library rely on two population-level variables describing the genetic composition of the population across generations:

First, the variable AlleleDiversity contains three 2-dimensional arrays (Figure A1.2).  The first table contains all possible alleles per locus of nuclear DNA. It consists of n lines, where n is the number of loci studied. Each row of the table contains a variable number of columns equal to the total number of potential alleles on the corresponding locus. Each cell in the array contains an integer, of type short, encoding the allele. This table is named nuclearAlleleDiversity.  The second and third tables are constructed according to the same principle. They correspond respectively to the cytoplasmic DNA of the mother and the father. These arrays are respectively called mCytoplasmicAlleleDiversity and pCytoplasmicAlleleDiversity.

When the haplotypes are alleles of a single locus, these arrays have only one line. Accessors : getAlleleDiversity (), getNuclearAlleleDiversity (), getMCytoplasmicAlleleDiversity (), getPCytoplasmicAlleleDiversity ().

Figure A1.2: Schematic representation of an AlleleDiversity

Second, the variable GeneticMap contains an array, called recombinationProbas, gathering probabilities of recombination between successive loci of nuclear DNA. It consists of n-1 lines, n being the number of loci studied on the nuclear DNA. Each line i contains a real, of double type, between 0 and 0.5, which represents the probability of recombination between the ith locus and the (i + 1)th. The order of the recombination probabilities in recombinationProbas respects the order of the loci as given in the tree genotype.

Both AlleleDiversity and GeneticMap have to be defined in the inventory file (see Appendix 2 below).

Population-level allele effects and heritability:

The different quantitative genetic parameters detailed in section 1.2 above are stored in a map called AlleleEffect whose keys are the parametric traits of interest (in our case FCRITBB and g1max). The map AlleleEffect contains as many sub-map as the number of parametric traits of interest (2 in our case). Each sub-map is named ParameterEffect, and contains three tables, and three parameters:  The first table (called nuclearAlleleEffect) contains on lines the loci of nuclear DNA that influence the corresponding parameter. The rows have a number of columns that are variable and equal to the number of possible alleles on the locus considered, denoted by n, +1. In each row, the first value is equal to the position of the locus in the array of the corresponding possible alleles (nuclearAlleleDiversity), that is, its row number. The n following values are the allelic effects (the{l}) of the n possible alleles of the locus. On each line, the sum of the n allelic effects of the alleles is equal to 0. In PDG, we assume that the loci coding for the studied traits are carried by nuclear DNA, so only nuclearAlleleEffect is not empty.  The second and third tables, respectively mCytoplasmicAlleleEffect and pCytoplasmicAlleleEffect, are constructed on the same principle and contain the loci of the cytoplasmic DNA, respectively maternal and paternal, which influence the parameter considered. In PDG, we assume that the loci coding for the studied traits are carried by nuclear DNA, so mCytoplasmicAlleleEffect and pCytoplasmicAlleleEffect are empty  The first parameter (real value of type "double") corresponds to the theoretical heritability of the parameter; this variable is called heritability. The second parameter corresponds to the total environmental variance, and is called totalEnvironmentalVariance. The third parameter corresponds to the proportion of the interStep-environmental variance in the total environmental variance of the parameter, and called interEnvironmentalVariance. These two last parameter aim at distinguishing a fixed source of environmental variation from a temporally-variable source of variation. In PDG, we assume most often that interEnvironmentalVariance = 0, so that totalEnvironmentalVariance= fixedEnvironmentalVariance Accessors : getAlleleEffect (), getParameterEffect (), getNuclearAlleleEffect (), getMCytoplasmicAlleleEffect (), getPCytoplasmicAlleleEffect (), getHeritability (), getTotalEnvironmentalVariance () et getInterEnvironmentalVariance ().

These variables must be defined in the inventory file (see example below in Appendix 2). Moreover, a main constraint of GENETICS library is that the allelic effects (the{l}) must be integer values. However, they are scaled by the value of the parametric trait, and the user may wish to have a finer precision than that of the parametric trait unit in the variation of allelic effect. Therefore, we introduced in PDG a parameter called AlleleEffectMultiplCoeff (typically set to 1000), which allow (i) to define in the inventory file the variable nuclearAlleleEffect with the required precision and (ii) to compute the genetic and phenotypic value of each individuals with the right scale of trait variation.

Individual genetic, environmental and phenotypic values

Moreover, each individual tree has got its own genetic, environmental and phenotypic values. These values are computed in PDG through the GENETICS library, based on the individualGenotypes and the AlleleEffect variables. They do not have to be provided, even for the initial stand. They are stored in maps of values, whose keys are the parametric traits of interest (in our case FCRITBB and g1max).

General hypothesis

We assume that the influence of environment on the variation at the parametric traits FcritBB and g1max is negligible, so that VE =0 ;VP=VA and h²=1. By contrast, the variation in the traits of interest themselves (TBB and WUE) is shaped both by genetic variation (through FcritBB or g1max) and environmental variation (through temperature or water stress).

General objective:

To create an inventory file, we need to control the initial additive variance VA and the allelic effects so that they match with the target distribution of each parameter at initiation. Moreover, we want to have an L-shaped distribution of the QTL effects.

Modeling variation at FcritBB in Fagus sylvatica

Assuming that the heritability of FcritBB is h² = 1, we consider that the value of FcritBB is determined by 10 bi-allelic and independent loci, with purely additive effects (see above).

In the initial population, we impose two constraints:  μFcritBB = 190 ° C, (ie the mean value of FcritBB known for Fagus sylvatica)  the coefficient of variation of FcritBB is 10% These constraints impose a target value for the standard deviation in FcritBB distribution: σFcritBB = 0.1 μFcritBB = 19.

Assuming that all loci contribute equally to the target additive variance VA = σ² = 361, the mean genetic value for locus l is : VA 1 = 36.1 (at each of the 10 loci). Moreover, the value of l corresponding to VAl depends on the allelic frequencies (Table A1.5). Therefore, to define the inventory file, we must simultaneously sample the allelic frequencies, the individual genotypes and the allelic effects l. This can be done using an iteration procedure implemented in a R script provided in: data/physiodemogenetics/generateInventoryFile/generateInvFile.R

Modeling variation at g1max in Fagus sylvatica

We follow the same strategy for g1max. Assuming that the heritability of g1max is h² = 1, we consider that the value of g1max is determined by 10 bi-allelic and independent loci, with purely additive effects (see above).

In the initial population, we impose two constraints:  g1max = 11, (ie the mean value of g1max known for Fagus sylvatica)  the coefficient of variation of g1max is 20% These constraints impose a target value for the standard deviation in g1max distribution: σ g1max = 0.2 μ g1max = 2.2

Assuming that all loci contribute equally to the target additive variance VA = σ² = 4.84, the mean genetic value for locus l is : VA1 = 0.484 (at each of the 10 loci). GENETICS library definitions Then the inventory file contains 3 sub-sections of parameters mandatory to initialize the GENETICS library.

The first sub-section is called here "#Genetic Map" and contains:  SpeciesName = a string of character giving the name of the species  speciesCode = an arbitrary integer  CastaneaCode = an integer linking the species to its code in the species parameter file for Castanea (here, 4 is for Fagus sylvatica)  The geneticMap of the species (see Appendix 1 section 2.2): here, the table of recombination probabilities is empty, meaning that all the loci are independent  The alleleDiversity of the species: here, only the nuclearAlleleDiversity is not empty, and contains 30 tables of two values (each coded as [1,2] and separated by ";"), corresponding to 30 bi-allelic SNPs (each value corresponding to the value of two possible alleles at this locus).

Note: empty tables (here, geneticMap, mCytoplasmicAlleleDiversity and pCytoplasmicAlleleDiversity) are coded by {}.

The second sub-section define the quantitative genetic parameters described in Appendix 1 section 1.2, with variables described in Appendix 1 section 2.3. This section contains one-line par focal parametric traits (here, two for FCRITBB and g1max):  speciesCode = an arbitrary integer matching the one given in section above  parameter = a string of character giving the name of the parametric trait  the allele effects ie the three tables nuclearAlleleEffect, mCytoplasmicAlleleEffect and pCytoplasmicAlleleEffect (see Appendix 1 section 2.3)  three values corresponding to heritability, totalEnvironmentalVariance and interEnvironmentalVariance (see Appendix 1 section 2.3)

The third sub-section is mandatory and defines the individual consanguinity of trees of the initial stand.

1.3.

Plot and cells definition The next section gives the list of cells with their properties … There is one line per cell, and each line (cell) contains:  An integer corresponding to the cell ID (cID)  Two integer corresponding to the coordinates on the y-axis (clign) and on the x-axis (ccol).

Note that the size of each cell in m is given in the general parameter above (in our case, each cell measures 50 m, and thus correspond to 0.25 ha). The total plot is thus 25 ha in area. Moreover, note that together with the parameters minimalAltitude and slopeInDegree (given in the general section above), these parameters define the altitude of each cell (see Figure A2.1 below for a plan of the simulated plot)  Two double values corresponding to the soil height (solHeight, in mm) and percentage of stone (stoneContent) at cell level. The soil Water Holding Capacity (WHC, in mm) is variable among cells and computed as: WHC = (1-stoneContent)×solHeight×(wfc-wallt)×Da where (wfc-wallt)=0.14 and Da= 1.07

In the inventory, the variability in WHC is artificially simulated by varying solHeight only (see Figure A2.2) Figure A2. 1: Simulated plot in the example inventory file. The grey grid corresponds to the 100 cells, with their cID indicated. In blue are given the coordinate of the center of the cell on the x-axis (bottom), on the y-axis (left side) and on the z-axis (right side).  ID: an integer from 1 to n, where n is the number of individuals (here n=2500)  cID : the cell ID to which the tree belongs  Species : an arbitrary integer matching the one given in GENETICS section above  rx : the relative x coordinate of the tree within the cell (in m)  ry: the relative y coordinate of the tree within the cell (in m)  DBH : Tree Diameter at Breast Height (in cm)  Height: Tree Height (in m)  Age : tree age; here, all trees are 40 years old, and will thus be considered as adult (as the general parameter adultAge=40)  nucDNA : a list of 60 integer corresponding to the genotype at the 30 locus of the nuclear DNA (the individuals are diploid, with one DNA inherited by their mother and the other by their father)  mCytDNA : here, an empty table  pCytDNA : here, an empty table  creationDate : the year of creation of the tree (here -1 indicate that this year is unknown)  mID : the ID of the mother of the tree (here -1 indicate that this ID is unknown)

Figure 1 :

 1 Figure 1: PDG-simulated relationship between each considered life history traits (A : Timing of Budburst, TBB; B: Water Use Efficiency, WUE) and the pertinent environmental variable (A : Temperature; B: Water stress index) for different genotypes (A : different FCRITBB; B: different g1max)

Figure 3 :

 3 Figure 3: Relationship between PDG data classes listed in Table2and other classes from the GENETICS library (in orange) or from the CASTANEA library (in green), of from CAPSIS kernel (in black)



  The map of an individual's genetic values contains the genetic (or genotypic) values of the different quantitative parameters studied. By definition, these values being invariable in time, the Map of the genetic values is also invariable in time. This variable is named genoValue.  The map of an individual's environmental values contains the fixed environmental values (representing the constant environmental effects) of the various parameters studied. The Environmental Values Map has been defined to allow the safeguarding of fixed environmental values as soon as they are calculated, these values being used for the calculation of the phenotypic values. This variable is named fixedEnvironmentalValue.  The map of the phenotypic values of an individual contains the phenotypic values of the different parameters at a given step of the evolution of the stand. The phenotypic value of a parameter is equal to the sum of the genetic value and the total environmental value (fixed part + variable part). Since an individual's phenotypic values are variable from Step to Step (since part of the environmental value varies from Step to Step), for a given individual, a Map of the phenotypic values can be calculated at each Step. This variable is named phenoValue.

Figure A1. 3 :

 3 Figure A1.3: Example of distribution of FcritBB and allelic effects

Figure A1. 4 :

 4 Figure A1.4: Example of distribution of g1max and allelic effects

  Target coefficient of variation of g1max used to draw trees of the initial stand

  Figure A2. 2: Variation of Water Holding Capacity (WHC, in mm) in the simulated plot

  "

	From To	mode Type	generation age
	0	20	g	Adult evolution without regeneration	G0	40-60
	20	30	r	Adult evolution with regeneration	G0	60-70
	30		i	Cut of G0
	30	60	s	Seedlings evolution	G1	10-40
	60	80	g	Adult evolution without regeneration	G1	40-60
	80	90	r	Adult evolution with regeneration	G1	60-70
	90		i	Cut of G1
	90	120	s	Seedlings evolution	G2	10-40
	120	140	g	Adult evolution without regeneration	G2	40-60
	140	150	r	Adult evolution with regeneration	G2	60-70
	150		i	Cut of G2
	150	180	s	Seedlings evolution	G3	10-40
	180	200	g	Adult evolution without regeneration	G3	40-60
	200	210	r	Adult evolution with regeneration	G3	60-70
	210		i	Cut of G3
	210	240	s	Seedlings evolution	G4	10-40
	240	260	g	Adult evolution without regeneration	G4	40-60
	260	270	r	Adult evolution with regeneration	G4	60-70
	270		i	Cut of G4
	270	300	s	Seedlings evolution	G5	10-40
	300	320	g	Adult evolution without regeneration	G5	40-60
	320	330	r	Adult evolution with regeneration	G5	60-70
	Figure

Table 2 :

 2 List of the 27 classes of the physiodemogenetics/model directory Color code : Data class [contains mainly object or parameters]; Process class [contains mainly methods corresponding to key ecological processes]; Tool data or process class [contains mostly data or methods implemented as tools for convenience]. Type : classical (in most model of CAPSIS), original (only in PDG), or Common (Common to demo-genetic tree model)

	Class	Type	Contents
	Functions	Original	Contains mathematical functions called by
			PDGSpread.java
	PDGCell	Classical	Defines the quadratic cell of a PDGplot and its
			methods; the PDGCell contain the PDGTree.
	PDGEvolutionParameters	Original	Embeds the evolution parameters, to be
			transmitted among classes (typically between
			script and model classes)
	PDGFmCell	Original	Defines the Castanea cell attached to each tree
			and its methods
	PDGFmGrowth	Original	Implements the methods for a tree to grow
			during a specified period of time using the
			Castanea library
	PDGGeneticOutput	Original	Implements the methods to export genetic
			results
	PDGGrower	Original	Abstract class defining the methods for the
			evolution step
	PDGGrowerAdult	Original	Defines the methods for the evolution step at
			adults stage without regeneration
	PDGGrowerAdultWithRegenerati	Original	Defines the methods for the evolution step at
	on		adults stage with regeneration
	PDGGrowerSeedling	Original	Defines the methods for the evolution step at
			seedlings stage (without regeneration)
	PDGInitialParameters	Classical	Contains parameters definition and value for
			PDG
	PDGLoiselleKinshipCoefficient	Common	Contains methods to compute spatial genetic
			structure within a tree population
	PDGMethodProvider	Common	Contains methods or call methods to compute
			spatial genetic structure within a tree population,
			or to compute stand characteristics
	PDGModel	Classical
	PDGModelLight	Classical
	PDGPlot	Classical	Define the physical map containing the
			PDGStand and its methods; the PDGPlot is
			made of several PDGcells
	PDGRapidEvolution	Original	Define the methods for pre-evolution
	PDGSimpleRegeneration	Original	Define the methods for the regeneration step
			(mating, seed + pollen dispersal)
	PDGSpecies	Classical	Define the genetic property attached to the
			considered population of each species.
	PDGSpread	Original	Define the methods for seed + pollen dispersal
	PDGStand	Classical	Define the stand (a collection of PDGTrees) and
			its method

  JT, Woodrow IE, Berry JA (1987) A Model Predicting Stomatal Conductance and its Contribution to the Control of Photosynthesis under Different Environmental Conditions. In: Progress in Photosynthesis Research (ed Biggins J), pp. 221-224. Springer Netherlands. Bost B, de Vienne D, Moreau L, Dillmann C, others (2001) Genetic and nongenetic bases for the L-shaped distribution of quantitative trait loci effects. Genetics, 157, 1773-1787. Dufrêne E, Davi H, François C et al. (2005) Modelling carbon and water cycles in a beech forest. Ecological Modelling, 185, 407-436. Falconer DS (1960) Introduction to quantitative genetics. New York,: Ronald Press Co. Oddou-Muratorio S, Davi H (2014) Simulating local adaptation to climate of forest trees with a Physio-Demo-Genetics model. Evolutionary Applications, 7, 453-467. Seinave I, Pichot C (2004) docGeneticsMars04_fc.pdf. INRA. http://capsis.cirad.fr/capsis/help_en/genetics?&#documentation Appendix 1: Parametrization of genotypes, allelic effects and heritability in PDG (following notations in Falconer 1960 and Bost et al. 2001)

Table A1 .1

 A1 Definition of the genetic value ml at a given locus lBost et al. (2001) consider a non-centered value, each locus l can contribute to a part ml of the trait TP, allelic effect l is an additive correction of locus contribution for the heterozygote (note: Bost et al.

	Genotype	Frequency	Value (Bost et al)
	Al1Al1	pl²	ml + l
	Al1Al2	2plql	ml
	Al2Al2	ql²	ml -l
	mean		ml + (pl -ql) l

Table A1

 A1

	.4 Summary of variables related to quantitative genetic parameter of interest and
	defined in CAPSIS
	Variable name in CAPSIS	Notation	Comment
	nuclearAlleleEffect	l }, table	NuclearAlleleEffect = l ×
		A1.2	AlleleEffectMultiplCoeff (because
			AlleleEffect must contain integer values)
			To be defined in the inventory file
	totalEnvironmentalVariance	VE eq A1.3 To be defined in the inventory file
	interEnvironmentalVariance	VE-IS	To be defined in the inventory file
	additiveVariance	VA eq A1.2 Not required in the inventory file
	heritability	h² eq A1.4 To be defined in the inventory file
	individualGenotypes		To be defined for the initial trees in the
			inventory file
	genoValue	A eq A1.1 Computed from individualGenotypes +
			AlleleEffect
	fixedEnvironmentalValue		Computed from AlleleEffect
	phenoValue		Computed from genoValue+
			fixedEnvironmentalValue
	3) Parametrization of allelic effects in PDG inventory file

Table A1 .

 A1 5 : Possible values of allelic effects corresponding to a single target VA (361) but with different allelic frequencies.

	pl	ql	²l	l	VAl	VA
	0.5	0.5	72.20	8.50	36.1	361
	0.6	0.4	75.21	8.67	36.1	361
	0.7	0.3	85.95	9.27	36.1	361
	0.8	0.2	112.81	10.62	36.1	361
	0.9	0.1	200.56	14.16	36.1	361
	0.95	0.05	380.00	19.49	36.1	361

 pID: the ID of the father of the tree (here -1 indicate that this ID is unknown)

R script to generate inventory file The R script generateInventoryFile_v3.R is available in the directory /data/physiodemogenetics/ to generate inventory files with the appropriate format. Basically, the script includes three parts: STEP1 = loading of parameters; all the parameters value are defined here STEP 2 = computations. Allelic effects for each trait are computed (optimization loop constrained by the target mean and standard variation of each trait) and individual's genotypes are drawn. STEP 3 = writing of the inventory file in the appropriate format Other results files to be improved and commented


In the domains of geography, urban development and spatial planning, graph theory is used to tackle questions arising in the field of networks (whether they are transport and communication or even social networks). The general procedure is based on the translation of these problems into the mathematical language of graph theory so as to be able to treat them using, for example, the properties of connectivity or by exploring minimum paths. For this reason, in its applications to the social sciences, graph theory deals mainly with mathematical properties and much less with the graphic representation of nodes and arcs 1 . By refocusing on the graphical dimension, from which it was born, here we wish to develop the aspect of graph theory that relates to representation (plotting of graphs) in its applications to problems of spatial analysis.

Since graph theory is particularly adapted to the modeling of transport networks, the representation of the network graph may constitute an investigative direction that can answer the difficult question of the representation of distances.

A classical illustration of the difficult problem of the representation of distances is provided by Müller [MUL 79,p. 216]. In Figure 9.2 we laid out three cities noted by p, q and r. There is a highway connecting the cities q and r, and another between q and p, whereas r and p are connected by a small road. The shortest path by duration between the two latter cities passes by the city q and follows the highway. The question raised by Müller is as follows: "How is it possible to determine, on a map with a time scale, the position of points p, q and r when the shortest path from one point to another is no longer a straight line?" [MUL 79, p. 215].

Small road Highway

.2. Representation of the shortest paths

1 When we study graph theory, one of the few properties that relates to their representationtheir plotting -is the concept of a planar graph. A graph is known as planar if it can be drawn on a plane so that the nodes are distinct points, the edges are simple curves and no two edges meet apart from at their ends.

The straight line almost never corresponds to the general form of movement, i.e., to the general form of geographical distance. Consequently, as Angel and Hyman put it: "if we wish to study transport within a geometrical framework, we must develop non-Euclidean geographical theories" [ANG 72,p. 366]. In this spirit we will start by establishing a general form of distance on a graph on the basis of the basic definitions of graph theory. The goal is to give a mathematical form to a geographical measurement: the distance built in this fashion will make it possible to find support in mathematics, to better envisage the spatial dimension associated with transport and movement and, finally, to question the cartographic representations used in geography.

To deal with the domain of representations we will create a method for reading distances on a map. The comparison of these measurements which are read with those provided by the geographical data will make it possible to characterize the various representations according to their capacity to represent geographical distances.

A distance on the graph

Mathematical distances are measurements between objects that respect the properties of positivity, uniqueness, separation and triangular inequality. From the mathematical point of view as well as from the point of view of geographical interpretation, one of the most fundamental properties of distances is that of triangular inequality because it introduces the idea of the minimum [LHO 97, p. 102 and 113]. The distance between two objects is associated with the measurement of the smallest gap that can be found. To illustrate this principle, on Müller's diagram the distance between points p and r can have two forms: if we measure it in kilometers, the distance is given by the small road, whereas, if we calculate it in duration of transport, the distance follows the highway route passing through point q.

In graph theory we have a particular application which associates a value to the shortest path: if a and b are two nodes of a graph G, the measure e (a, b) is given by the length of the shortest path between a and b.

If we consider the case of a valuated graph 2 , the length of a path corresponds to the sum of the values associated with the arcs, of which it consists. With the measure e, we construct an application that associates a value in R to a pair of elements of the set of nodes S. It should be noted that in this definition there is no specification of the mathematical properties of the application. In particular, it is not specified, if the measure e is metric, i.e., if it respects the properties of distances.

In the general case of a non-valuated graph Flament indicates the construction of a metric on the basis of the measure e [FLA 68, p. 39]. The only condition for the graph is that the value associated with a link between two nodes is identical in each direction. The graph must therefore be symmetrical. The measure between one point and another is considered zero. Defined on a symmetrical non-valuated graph, with conventions for infinite and zero paths, the margin is metric.

On a valuated graph in order for the measure e to be a distance in the mathematical sense, we must verify one by one the metric properties (symmetry, positivity, separation and triangular inequality) for any pair of nodes. That leads to two conditions, which make it possible to establish a graph distance [LHO 97, p. 137]: the graph must be symmetrical and for a valuated graph the values associated with the arcs must be strictly positive.

The distance criterion of minimum paths calculation can integrate the duration of transport, the cost, the length of the journeys, but also the various modes of transport (with a quality of connection that may or may not be homogenous) or even the nature of what is transported. If the graph considered is not symmetrical, we can construct only a non-symmetrical metric, a structure that belongs to the more general field of impoverished metrics3 . Although non-symmetry is the general case for geographical distances (in time or cost) we will limit ourselves here to the analysis of symmetrical distances by considering a simplification of the collected data.

The distance defined in this fashion is purely mathematical: it is only the connection between pairs of places with a measurement that takes the form of a measure e. It is a mathematical object that characterizes geographical data, but a priori does not have immediate and univocal translation in the sphere of cartographic representations.

A distance on the map

Although a geographical map primarily provides information about the location of places, it also informs us about the relations between these locations. In particular, reading a roadmap makes it possible to work out routes and determine the distances associated with them.

To define the function of distance that we have in a map Müller [MUL 79,p. 224] introduced the concept of "graphic distance". This distance is defined in an Euclidean form: we consider that the length of a path between two points on the map is directly proportional to the length of the rectilinear segment which connects them. However, movements carried out in the geographical space use transport infrastructures that have very little chance to follow a straight line between two places chosen at random. Thus, in order to measure a distance in kilometers on a roadmap the reader must transform the graphic distance mentally to obtain a useful measurement: the sinuosity of the road and the extent of detours along the way make it possible to correct, revising up, the Euclidean length of a route. In this context the Euclidean distance offers a first approximation of measurement. It is then adjusted by information about the infrastructure used. In fact, the concept of distance "as the crow flies", on which Müller's graphic distance is based, cannot be used to read a cartographic representation if we wish to understand the logic of movements.

For this reason it is necessary to introduce a different concept, that of "visual length" [LHO 97, p. 137], which is derived from Müller's graphic distance. The visual length of a connection between two points is equal to the length of the corresponding path. If, for example, the path takes the form of several segments placed end-to-end, the visual length corresponds to the sum of the lengths of these segments. Visual length is, thus, not an Euclidean concept. It should be noted that, according to this definition, visual length does not exist between two points unless there is a path. Visual length makes it possible, in particular, to provide a measurement of the routes and paths on the basis of a graph plot, i.e., on the basis of the map of a transport network.

By abandoning graphic distance in favor of visual length we lose the reference to the metric. The motivation of this choice becomes apparent when we refer to Müller's diagram (Figure 9.2). The visual length of the connection between p and q corresponds to the distance between p and q, if it is measured in kilometers. However, that is not true if the distance is calculated in journey duration. In this case the visual length read on the map along the small road is not a distance since it does not correspond to the minimum possible measurement in space-time.

By uncoupling the concepts of distance (geographical concept) and visual length (cartographic concept) we pave the way for a better understanding of the representations of distance.

Thusly defined, the concept of visual length calls into question the usual use of map scale. Indeed, on a traditional geographical map the scale has the function to provide measurements of distance in kilometers as the crow flies: it is clearly an Euclidean notion of distance. This use of scale implies the postulate of verifying the isotropy and the uniformity of the areas represented: two properties that cannot be regarded as given when geographical spaces are dealt with.

Visual length establishes a measurement which, converted by using the scale of representation, produces a geographical distance. It is a non-Euclidean use of the map scale. A cartographic representation of distances is considered to be satisfactory, if the distances read are coherent with the initial data, that is, with geographical distances.

For Bunge there are two manners of representing distances: we may show on a traditional map the paths that can be sinuous, or we may represent distances in a simplified form but on a deformed map [BUN 66]. The second approach refers to the anamorphoses introduced by Tobler [TOB 63, p. 59-78] and consists of moving the locations of places according to distances4 . This second approach is the one that interests us because it directly involves the representation of the transport network.

Spring maps

To illustrate the impact of the modes of transport on spatial relations Tobler considers the representation of distances in a mountainous zone [TOB 97]. The initial data set consists of measurements of real distances in kilometers that often follow sinuous routes. The degree of circuity of the associated graph (which is defined as the measurement of the difference between the distances in kilometers and the distances as the crow flies [KAN 63, p. 93-121]) is thus very high5 . The spring map (Figure 9.3) is constructed on the basis of a set of places which are positioned at their real location [CAU 84, p. 40], i.e., their position on the plane of the map is not modified by anamorphoses.

The principle of spring representation is to depict the link existing between two points in space in the shape of a spring whose length is proportional to the length in kilometers. By reference to the Euclidean distance, connections which are longer than those "as the crow flies" are represented in the shape of a generally compressed spring. This graphic principle makes it possible to represent paths which are longer than the straight line without modifying the location of the ends.

To read such a map, it is necessary to get rid of our Euclidean reflexes because the distance between places is indicated by a path which is not a straight line. This distance can be understood only by using visual length because the graphic distance defined by Müller results in Euclidean measurements. The map is read by evaluating the intensity of the compression of the springs: the more compressed the spring is, the more difficult the connection (in other words it contains more turns and slopes due to the mountainous terrain, which lowers the average speed) and the more it moves away from the straight line. The representation is built around a direct reference to the Euclidean distance, but a reference that is implicit. Here we find the status of the Euclidean space as proposed by Cauvin, that is, as a reference space [CAU 84, p. 72]. The spring graph may be seen as the plotting of a graph. The graph is that of the transport network considered, which is valuated by the measurements taken for the network, and its plotting undergoes three constraints:

-the nodes of the graph are placed at the geographical position of the nodes, which they model; -the length of the arcs of the graph is proportional to the associated value; -the arcs of the graph are drawn in the shape of springs which ensure the proportionality of the lengths.

The distances in kilometers are considered here to be symmetrical and they define a graph distance. The construction principle of the spring map makes it possible to establish a representation which is coherent with the associated mathematical structure. Therein lies the main merit of this type of map, which is to offer a representation of distances that remains coherent with real measurements, although they elude the Euclidean domain.

On the basis of the measurements represented on the spring map, Tobler then proposes three-dimensional anamorphoses. The adopted principle is to slide the location of the places on a vertical axis to stretch the distances "as the crow flies" in three dimensions. A regression using least squares makes it possible to find a configuration which minimizes the cumulated error for all the links (Figure 9.4.). Then, a transport surface can be calculated by interpolation, by building a continuum based on the previously positioned sites (Figure 9.5). It is possible to find the distances between places by using a measurement along a two-dimensional plane. Let us imagine tracing the original transport network on this surface. This plotting of the graph would make it possible to find a coherence between the real distances and the represented distances, similarly to what is proposed by the spring map. If we consider that the initial data makes it possible to construct a graph distance (where positivity and symmetry are verified), the suggested map is coherent with this data, as well as with the mathematical structure that they imply and therefore also with the geographical reality.

The construction principle of the interpolated transport surface comprises, however, two important differences with respect to the spring map representation mode. Firstly, regression and interpolation introduce an error, which, even if minimized, makes the representation enter the domain of approximation, since the measurements taken have to integrate an associated error. Secondly, the use of the third dimension implies a projection system onto a two-dimensional surface of a paper sheet that carries the image. And there too we introduce an approximation, though of a different nature, due to the projection angles. This approximation can be largely reduced, if we help the reader to reconstruct the third dimension mentally. In Figure 9.7 the deformation of the grid, the hidden surfaces (in the south of the surface) and the three-dimensional coordinates system invite the reader to enter into the third dimension of the map and help him to reconstruct the surface topology. The second application that we propose [PLA 87, p. 12] (Figure 9.6) shows the quality of service of the rail network in the Lyon region. The value of the arcs in the graph indicates the quality of service, which in turn is constructed on the basis of the quality of service of every line.

Visually, on spring maps, it is difficult to directly perceive the exact length of the arcs. We first see the generally "compressed" aspect of the spring which provides an indication of its length. The notion of visual length, which is given by the overall length of the spring, does not result here from a direct visual perception. In the case of the spring graph we are closer to the idea of a mental transformation carried out by the map reader, than of a direct and proportional visual perception. However, does this mental transformation not proceed from the learning inherent to any nontraditional system of representation? As Tobler proposes, these representations may seem strange at first, but this is largely due to our more favorable attitude towards more traditional and also more familiar maps, which means that we tend to consider only conventional maps as realistic and correct [TOB 61, p. 164].

To answer the problem of the representation of distances, the plotting of a threedimensional graph offers other latitudes and allows other modes of representation, which are to some extent connected with Tobler's explorations. We will now present the space-time relief maps, from the viewpoint of coherence with the initial data, but also by way of comparison with the systems presented up until now.

ChronoMaps: space-time relief maps

Space-time relief maps, or chronomaps, aim at representing a space of transport deformed by the coexistence of modes of transport with different performances [LHO 96, p. 37-43]. In the domain of passenger transport speed and cost per kilometer vary very strongly according to the transport modes, but also within a modes depending on the quality and type of the infrastructure used. An inhabitant of Aurillac who wishes to go by car to the prefecture of the area, Clermont-Ferrand, starts by driving on a trunk road with an average speed of 60 km/h until he reaches a highway entry, from where he drives at 110 km/h on average. In this example, the highway portion of the journey is covered almost twice as quickly as the portion on the trunk road. The possibility of swapping between the sub-networks of the transport system are very strongly conditioned by the nature of the networks and, therefore, the cartography of such a transport space must take into account the form of the networks and the heterogeneous nature of the system. This difficult problem, which in anamorphoses is dealt with by dilating the space in a non-homogenous manner according to the quality of the connections, is solved on the relief map by employing the third dimension. To represent these distances, we do not change the location of the places, but the way of drawing the connections.

Chronomaps are constructed on the basis of a graph (Figure 9.7 of the central book, stage b). The graph representing the transport network must be planar and saturated in order to enable the construction of a three-dimensional surface (Figure 9.7, stage g). To respect the heterogeneity of the network the graph adopts the shape of a p-graph6 , in which each sub-network appears as a partial graph7 . Thus, between Nantes and Angers, the highway and the road are superimposed in the p-graph and each infrastructural network is considered as a partial graph of the p-graph. The nodes of the graph are again assigned to the main cities of the area considered, but also to the singular places, whose consideration is necessary for a thorough understanding of the transport network, such as the Essarts highway interchange located between La Roche-sur-Yon and Cholet. The choice of the density of nodes in the space considered is one of the determining elements of the form of the relief and, thus, of the produced image and the message that it conveys. Low density can make it possible to show the effect of networks with a broad grid (TGV and highway) over the entire area, whereas greater density makes it possible to express the complexity of local distortions.

Representing distances in a homogenous network with connections that are not far from the straight line is easy: in the plot in the plan of the highway graph (stage c) the length of the arcs plotted is directly proportional to the time-distance and the associated errors are small. On the other hand, in a heterogeneous network, where the differential journey speeds (or costs) are considerable, the proportionality of the lengths of the arcs cannot be respected if they are drawn in the form of segments in the plan. In space-time relief maps the arcs with poorer performances are traced under the plane of the nodes in the shape of two segments, in such a way that their length is proportional to the journey durations. The worse a connection performs, the more an arc becomes concave and moves away from the straight line on the plane (stages d, e and f). During the last stage the coloring of the graph surfaces in different shades of gray makes it possible to reconstruct the transport surface and facilitates reading the relief.

The chronomap is constructed as the plotting of a graph representing a multimodal transport network. The principle of plotting follows three rules that can be stated as follows:

-the position of the nodes in the graph corresponds to their real position; -the length of the arcs in the graph is proportional to their effective length (be it in kilometers, hours, etc); -to enable the proportionality of the lengths, the arcs are drawn in the third dimension.

From the point of view of graph plotting, the relationship of this representation principle with that of Tobler's spring maps is very strong. The location of the nodes is the same and the principle of proportionality for the length of the arcs is common. The only difference lies in the manner of tracing the arcs.

Another interesting point of comparison of the two models is their way of representing the continuum. According to Tobler, the passage from graph plotting to the transport surface is achieved by using a regression along a vertical axis followed by interpolation. In the space-time relief map the passage from graph plotting is established by tracing the facets of the graph on the basis of the arcs. Each triangular facet (the graph must be planar and saturated) is in turn divided into four triangles which are traced from the middle of the facets. The surface constructed in this manner then undergoes the deformation of graph arcs. The surface is associated with the road graph and not to the highway graph because access to normal space is only possible by using the road. This idea of universality of the road network [DUP 94, p. 146], defined by Dupuy, is expressed by Tobler by defining the border of the network: the border of the road network is constituted by the edge of the carriageway, whereas the border of the highway network is constituted by the access points [TOB 61, p. 89]. On the relief map the slope of the arcs directly translates the associated relative speed: the more marked the slope, the lower the connection speed is with respect to the possible maximum speed appearing in the plan of the nodes occupied by the highway network. By geometrical construction the slope of the facets in the graph is sharper than that of the arcs upon which they are based. This means that the performance of the interstitial sub-network that services the area of the facet is poorer than that of the connections bordering it. This principle is coherent with that of network hierarchy: in transport networks the performance drops with the size of the grid [PLA 91, p. 22]. The representation of the continuum, which is defined as the dual of the network, does not contradict the principle of relief representation. However, it is necessary to draw attention to the fact that the distances shown by the relief map are, first of all, those produced by the transport network. Here, the first function of the continuum is graphic: the shades of gray of the facets help the reader to imagine the third dimension, just as the deformed grid used by Tobler did.

Chronomaps do not show space in the form of plane layers, but of superimposed complex surfaces. The image in Figure 9.8 of the central book shows two subnetworks of the road transport network which are arranged in superimposed layers in the third dimension and coming into contact only at the sites of highway interchanges. This is a direct illustration of the tunnel effect of fast infrastructures.

Each space is registered according to its own relief: the highway is in the plane of the cities where the straight line draws the fastest path but where space is reduced to a set of points (network space), whereas the other networks follow a relief which becomes more marked as accessibility diminishes (banal space). The space of the road is a continuous but irregular surface, less accessible, but where the concept of proximity still operates. The map shows a space dualized by speed.

The large highway axes which serve the territory appear clearly inscribed in the plane of the cities. However, in the interstices of this network the space-time relief reflects the lack of accessibility due to the inferior quality of the infrastructure. The network space of the Riviera, from Avignon to Menton, which is well connected thanks to an extensive highway grid, strongly contrasts with the poorly accessible space-time of Alpes-de-Haute-Provence and Hautes-Alpes.

The ChronoMap of the road and the highway renders readable the minimum duration transport routes by showing the degradation of the relative conditions of circulation outside of the highway type network. Thus, the Grenoble-Nice connection is ensured via the Rhone valley along a highway route that traces a vast loop whose visual length remains lower than that of the route closer to the straight line through Gap and Digne which passes through an irregular space-time relief.

In formalizing the construction principle of relief maps, the first constraint relates to the length of the arcs without taking their visual length into consideration. In this respect we may note a difference with the construction principle of "spring maps". In spring maps visual length is directly proportional to the effective length of the arcs, even though the visual length of springs is more inferred than directly read.

Relief maps are particular views of a general three-dimensional model. The length of the arcs of the construction principle is a constraint on this general model. The images, whether projected onto a screen or onto a sheet of paper, are twodimensional. The concept of visual length may be used only with respect to the images. An image is created by the transformation of a three-dimensional structurethe model -into a two-dimensional form -the map. Just as on the interpolated transport surface of Tobler, the complex play of three-dimensional structure projections onto the plane of the map causes disturbances in the lengths of the arcs. These disturbances justify the fact that we only have an "approximate scale" of space-time in relief maps. The visual length which is available for images is not exactly proportional to the effective length present in the data. Mathematically, this results in the fact that a three-dimensional surface and the final image are not isometric: there is a distortion due to the projection on the plane. For arcs belonging to the third dimension, the distortion between visual lengths and effective lengths is not homogenous. It is therefore difficult to evaluate visually: it is necessary to find the position of the segments in three dimensions. It is the role of gray shaded facets, of viewing angle indication, to help to mentally reconstruct the relief in order to make the distortion understandable. Let us note that the general principle of relief maps construction places no constraints on images, meaning that the adjustment of the image generation parameters arises from the choices made by the cartographer. An infinite number of possible images correspond to the single model, according to the choice of the viewing angle, or also of the point of view8 .

Although the approximation due to the projection of the three-dimensional structure is common to that of the interpolated transport surfaces of Tobler, quite unlike the latter, from the point of view of the construction principle, space-time relief maps address the exact domain. This means that the length of arcs of the threedimensional model is strictly equal to that coming from initial data.

Conclusion

Distance belongs to concepts applied in disciplines that are very far from each other. In Tobler (for whom any map results from a mathematical transformation whose parameters it suffices to modify), in Müller [MUL 79,, or in Huriot and Perreur [HUR 90,p. 225] the issue of the representation of distances, in geography as well as in other disciplines that concern space, insistently refers to mathematics. In this spirit, the representation procedures exposed here are all based on graphs. The very object of this contribution (the issue of the representation of distances) is a call for a positioning in the field of metrics. Graph distance has been defined in this manner. The representations all are constructed on the basis of a graph plotting. Reading the map requires measuring instruments (which are the scale of representation and the graphic length), which make it possible to reconstruct metric space and understand its distance.

Whether implicit or explicit, the reference to mathematical structures is necessary for the procedure of representing distances (Figure 9.9). The use of mathematical forms enables the clarification of principles and ensures the freedom of contradiction [BUN 66, p. 2].

The principles of representation clarified here show the essential contributions of graphs use in the cartography of distances and the understanding of spatial relations. All the avenues are far from having been explored and the models still hold important development prospects. Indeed, if the relief maps presented show the connections in the form of two broken segments, nothing prevents us from imagining other configurations using curves, for example.

The cartographies of distance can hardly be understood without instructions (a scale of correspondence and the way to use it), which sometimes disturb the established order of cartographic conventions.

Figure 9 . 3 .

 93 Figure 9.3. Spring graph of the road network of Western Colorado

Figure 9 . 4 .

 94 Figure 9.4. Road distances approximated by displacing the nodes in the third dimension

Figure 9 . 5 .

 95 Figure 9.5. Transport surface constructed by interpolation

Figure 9 . 6 .

 96 Figure 9.6. "Spring" graph of the Lyon region

Figure 9 . 9 .

 99 Figure 9.9. The role of mathematical structures in the representation of distances

In a valuation graph a numerical value is associated to each arc.

For more on these structures and their use in spatial analysis see[HUR 90, p. 225].

On this subject see [CAU 96] (www.cybergeo.presse.fr).

For developments and analyses of the degree of circuity see[CHA 97, p. 330].

In a p-graph there can be no more than p distinct arcs connecting the same pair of nodes.

A partial graph G' of a graph G contains all of the latter's nodes, but only a part of its arcs.

For relief maps of the Atlantic coast according to an unusual point of view see [MAT

96, p. 97-111].