Additive genetic variance (*VA*) and total genetic variance (*VG*) room core principles in biomedical, evolutionary and production-biology genetics. What identify the large variation in report *VA*/*VG* ratios indigenous line-cross experiments is no well understood. Here we report exactly how the *VA*/*VG* ratio, and also thus the ratio between narrow and wide sense heritability (*h2*/*H2*), varies as a function of the regulatory style underlying genotype-to-phenotype (GP) maps. We studied five dynamic models (of the cAMP pathway, the glycolysis, the circadian rhythms, the cabinet cycle, and heart cell dynamics). We assumed hereditary variation to be reflected in model parameters and also extracted phenotypes summarizing the device dynamics. Even when imposing purely linear genotype come parameter maps and also no eco-friendly variation, us observed quite low *VA*/*VG* ratios. In particular, solution with optimistic feedback and cyclic dynamics gave much more non-monotone genotype-phenotype maps and also much reduced *VA*/*VG* ratios than those without. The results show that part regulatory architectures consistently preserve a transparent genotype-to-phenotype relationship, whereas various other architectures generate more subtle patterns. Our method can be offered to elucidate this relationships throughout a whole range of organic systems in a systematic fashion.

You are watching: Broad sense vs narrow sense heritability

The *broad-sense heritability* the a properties is the proportion of phenotypic variance attributable to genetic causes, when the *narrow-sense heritability* is the relationship attributable to additive gene effects. A much better understanding the what underlies sport in the ratio of the 2 heritability measures, or the identical ratio of additive variance *VA* to complete genetic variance *VG*, is important for production biology, biomedicine and evolution. We find that reported *VA*/*VG* values from line crosses vary greatly and also ask if organic mechanisms basic such differences can be elucidated by linking computational biological models v genetics. To this end, we made usage of models that the cAMP pathway, the glycolysis, circadian rhythms, the cabinet cycle and also cardiocyte dynamics. We assumed additive gene action from genotypes to version parameters and studied the result GP maps and *VA*/*VG* ratios of system-level phenotypes. Our results present that some varieties of regulation architectures consistently keep a transparent genotype-to-phenotype relationship, whereas others generate an ext subtle patterns. Particularly, systems with hopeful feedback and cyclic dynamics resulted in more non-monotonicity in the GP map bring about lower *VA*/*VG* ratios. Our technique can be supplied to elucidate the *VA*/*VG* relationship across a whole selection of organic systems in a methodical fashion.

**Citation: **Wang Y, Vik JO, Omholt SW, Gjuvsland abdominal (2013) effect of Regulatory design on broad versus Narrow feeling Heritability. jajalger2018.org Comput Biol 9(5): e1003053. Https://doi.org/10.1371/journal.pcbi.1003053

**Editor: **William Stafford Noble, university of Washington, United states of America

**Received: **August 10, 2012; **Accepted: **March 23, 2013; **Published: ** might 9, 2013

**Funding: **This work was sustained by the research study Council that Norway (http://www.rcn.no) under the eVITA program, task number 178901/V30, and by the digital Physiological Rat job (http://virtualrat.org) funded through NIH approve P50-GM094503. The advancement of the cgptoolbox was supported by the VPH-Network the Excellence (http://vph-noe.eu) v exemplar project EP7. NOTUR, the Norwegian metacenter because that computational science, noted computing sources under job nn4653k. The funders had no role in study design, data collection and also analysis, decision come publish, or ready of the manuscript.

**Competing interests: ** The authors have claimed that no completing interests exist.

## Introduction

The broad-sense heritability of a trait,

, is the relationship of phenotypic variance attributable to genetic causes, when the narrow-sense heritability , is the ratio attributable to additive gene action. The total genetic variance consists of the variance defined by intra-locus supremacy () and inter-locus interaction (). The factors for and also importance the this non-additive genetic variance that distinguishes the two heritability measures has been subject to substantial controversy for more than 80 year (e.g., <1>–<6>). It to be recently displayed through statistical debates that for traits with countless loci at excessive allele frequencies, lot of the genetic variance becomes additive with*h2*/

*H2*(or equivalently

*VA*/

*VG*) generally >0.5 <3>. In populaces with intermediate allele frequencies, together as managed line crosses, lower

*VA*/

*VG*ratios are frequently reported <7>, <8>. This is illustrated in Table 1, i m sorry summarizes approximated

*VA*/

*VG*ratios indigenous a collection of studies on such populations. This wide range of

*h2*/

*H2*ratios reported for line crosses cannot be described by an allele-frequency argument, and also putative explanations should be based on how the regulatory design of the underlying organic systems form the genotype-phenotype (GP) map.

Download:

Table 1. examples of report

*VA*/

*VG*ratios of indigenous line-crossing experiments.

https://doi.org/10.1371/journal.pcbi.1003053.t001

It is important to understand the causal underpinnings that the observed variation in *h2/H2* ratios within and also between organic systems for numerous reasons. In person quantitative genetics, whereby twin studies are typically used, many heritability approximates refer to *H2* <9>. In cases where *h2/H2* is low, this deserve to lead to unrealistic expectations around how much of the underlying causative variation may be located by direct QTL detection approaches <6>. On the various other hand, low narrow sense heritability for a given complex trait does not necessarily indicate that the atmosphere determines lot of the variation. In evolutionary biology, additive variance is the foremost currency for evolution adaptation and also evolvability. Vital questions in this context are for instance (i) to which level is there an option on the regulation anatomies us to preserve high additive variance, (ii) are there business constraints in building adaptive solution such the in some situations a low *h2*/*H2* ratio have to of necessity emerge while the proximal equipment is quiet selected for? Moreover, in production biology with genetically modified, sexually reproducing organisms, one would choose to ensure that the changes would be passed over to future generations in a totally predictable way. Thus, one would prefer to ensure that the modification becomes extremely heritable in the narrow sense.

As a action towards a physiologically grounded knowledge of the sports of the *h2*/*H2* relationship throughout biological equipment or processes, us posed the question: are there regulation structures, or details classes that phenotypes, more likely to create low *VA*/*VG* ratios than others? Addressing this question requires the linking of hereditary variation to computational biologic in a population context (e.g., <10>–<19>), so-called causally-cohesive genotype-phenotype (cGP) modeling <15>, <17>, <18>. We used this strategy to 5 well-validated computational biologic models describing, respectively, the glycolysis metabolic pathway in budding yeast <20>, the cyclic adenosene monophosphate (cAMP) signaling pathway in budding yeast <21>, the cell cycle regulation the budding yeast <22>, the gene network underlying mammalian circadian rhythms <23>, and also the ion channels determining the activity potential in computer mouse heart myocytes <24> these models differ in their regulatory architecture; below, we display that they additionally differ in the range of *VA*/*VG* ratios the they can exhibit. In particular, hopeful feedback regulation and oscillatory behavior seem come dispose for low *VA*/*VG* ratios. The results indicate that our approach can be used in a share manner to probe exactly how the *h2*/*H2* proportion varies together a function of regulation anatomy.

### Simulations of cGP models

The five cGP models were built and analyzed with the *cgptoolbox* (http://github.com/jonovik/cgptoolbox) an open-source Python package arisen by the authors; further source code details to the simulations in this document is accessible on request. In the following we explain the three key parts the the workflow: (i) the mapping from genotypes come parameters, (ii) the mapping from parameters to phenotypes, i.e. Fixing the dynamic models and also (iii) the setup the Monte-Carlo simulations combine the 2 mappings (Figure S1). Because that each model, we briefly describe its origins, the software offered to deal with it, i beg your pardon parameters were subject to hereditary variation, what phenotypes were recorded, and also criteria for omitting outlying datasets. Figures S2, S3, S4, S5, S6 reflects graphical representations of the 5 model systems and also Text S1 contains much more detailed descriptions of all five models.

Genotype to parameter mapping.

For each model, the adhering to procedure was recurring 1000 times (see also “Monte Carlo simulations” below) for different selections that parameters to be based on simulated genetic variation. We began by sampling 3 polymorphic loci, every determining one or 2 parameters in the dynamic model. Tables of standard loci with matching parameters and their baseline worths are provided in Table S1, S2, S3, S4, S5, equivalent to the cAMP, glycolysis, cabinet cycle, circadian and action potential models respectively. Heritable variation in a liked parameter was generated for a single biallelic locus v allele indexes 0 and 1 in the complying with manner. First, two numbers *r1* and *r2* were sampled uniformly in the interval <0.7, 1.3>. The parameter worth for a homozygote 00 was set to

*b*is the baseline value, because that a homozygote 11 the parameter worth was . The heterozygous genotype 01 was assigned the mean of the 2 homozygotes , leading to an additive mapping native genotypes to parameter values.

cAMP model.

Graphical illustration that the phenotypes tape-recorded for the 5 cGP models studied. Time courses (state variable on y-axis, time ~ above x-axis) for the baseline parameter collection are presented for all five models. **A.** In the absence of exterior glucose all state variables (concentration of cAMP is shown) in the cAMP design <21> converge come a steady steady state (blue circle on y-axis). After addition of external glucose (5 mM included at time 50) we view a rapid change followed through a slow return to the initial steady state. In addition to the original steady state, the extremal concentration (top blue circle) and the time to with the extremum (blue line) was taped as phenotypes. **B.** Metabolite concentrations (internal glucose (GLCi), glucose-6-phospate (G6P) and also fructose-6-phospate (F6P) are shown) in the glycolysis version <20> every converge come a steady steady state, indicated by open up circles. The steady state concentrations because that 13 metabolites were tape-recorded as phenotypes native this model. **C.** for the cabinet cycle version <22> we recorded the height level and also the time native bottom to optimal as because that the circadian design (Figure 1D), and also in addition we tape-recorded cell cycle events such together bud introduction at the time once **D.** mRNA and also protein concentrations (mRNA because that *Bmal1* (MB), *Cry* (MC) and *Per* (MP) space shown) in the circadian version <23> converge come a limit cycle. In addition to the duration of oscillation (long blue line) for each of the 16 variables the optimal level (open blue circle) as well as the time native bottom to peak (short blue line) were recorded as phenotypes. **E.** We offered the base level, optimal level, amplitude, time come peak, and also time to 25%, 50%, 75% and 90% restore of the activity potential and calcium transient as cell level phenotypes that the action potential design <24>. An action potential is shown in the figure.

https://doi.org/10.1371/journal.pcbi.1003053.g001

Glycolysis model.

The design published by Teusink et al. <20> describes glycolysis in *S. Cerevisiae* with the kinetics the 13 glycolytic enzyme determining the fluxes that metabolite state variables. Genetic variation was presented on maximal reaction rates for the enzymes (see number S3 and Table S2). Us downloaded the design from the BioModels database (http://www.ebi.ac.uk/biomodels-main/BIOMD0000000064) in SBML L2 V1, and solved it v PySCeS <25> to uncover the stable steady state concentration of metabolites, which were used as phenotypes (see number 1B and also Table S7). Datasets to be discarded if one or an ext of the genotypes did not offer a stable steady state, as have the right to happen because of a saddle-node bifurcation <26>.

Cell bike model.

The model of the agreement control instrument of the cell cycle in *S. Cerevisae* modeled by algebraic/differential equations that define the constant changes in state variables and also discontinuities because of cellular occasions <22> was acquired from the CellML repository (http://models.cellml.org/workspace/chen_calzone_csikasznagy_cross_novak_tyson_2004). Hereditary variation was introduced on the production and also decay rates of assorted proteins (see number S4 and also Table S3). The released model has reset rule (events) for both parameters and also state variables, yet the CellML implementation only includes the parameter (*k*mad2, *k*bub2 and *k*lte1) rules. Reset rules because that state variables

Circadian model.

The design of the mammalian circadian clock published by Leloup and also Goldbeter <23> explains the dynamics that mRNA and also proteins of three genes in the cytosol and also nucleus. Hereditary variation was presented on mRNA degeneration rates (see figure S5 and also Table S4). The design was downloaded from CellML repository (http://models.cellml.org/workspace/leloup_goldbeter_2004) and also integrated using CVODE <27> until convergence to its border cycle. Together phenotypes we offered the bottom levels and also time to from bottom level to peak value the the concentration of mRNAs, proteins and also protein complexes. In addition, we videotaped the duration of oscillations (see number 1D for phenotype illustrations and Table S9 for phenotype descriptions).

Action potential model.

The design of <24> is an extension of <28> and also describes the activity potential and also calcium transient that a mouse heart muscle cell. We derived CellML code from the authors and also the document is easily accessible as supplementary material in <17>. Number integration was done making use of CVODE <27>. Hereditary variation was presented on the maximal conductances that ion channels and also pump six (see figure S6 and also Table S5). Phenotypes were created by simulated constant pacing together done in <17>, <18>, through a economic stimulation potassium present of −15 V/s to be lasting because that 3 ms in ~ the start of each stimulus interval. The version was simulated to convergence as described in <17>; datasets that failed to converge were discarded. The early stage level, optimal level, amplitude, and also time to 25, 50, 75 and also 90% recovery of the action potential and also calcium transient were taped as the cabinet level phenotypes (see number 1E because that phenotype illustrations and also Table S10 for phenotype descriptions).

Monte Carlo simulations.

For each design we carry out 1000 Monte Carlo simulations as complies with (see number S1 for an illustration). We very first sampled three polymorphic loci for advent of genetic variation and also sampled the genotype-to-parameter map as described above. Then all 27 possible three-locus genotypes were enumerated, mapped into 27 parameter sets and also for every parameter set the dynamic version was solved and phenotypes (as described over and in figure 1) were obtained. To protect against artifacts emerging from number noise datasets with low variability were omitted native the hereditary analysis. Pure variability to be measured together the difference between the maximum and also minimum values of a phenotype in a dataset, and relative variability together the proportion of the pure variation come the average phenotype the the very same dataset. The threshold values for each phenotype and also the number of datasets exceeding the thresholds are detailed in Tables S6, S7, S8, S9, S10, because that the cAMP, glycolysis, cell cycle, circadian and activity potential models, respectively.

Decomposition of hereditary variance.

A single Monte Carlo simulation results in genotype-to-phenotype maps consisted of by 27 genotypic worths (i.e. The phenotype values matching to the 27 genotypes) because that a given phenotype. We offered the NOIA structure <29> come compute the resulting genetic variance (

) in a hypothetical F2 populace and decompose it into additive () and also non-additive materials ( and ). This to be done v the function*linearGPmapanalysis*in the R parcel

*noia*(http://cran.r-project.org/web/packages/noia/) variation 0.94.1.

Monotonicity that GP-maps.

We construct on the definitions of monotonicity and also the indexing that alleles presented in <30>. Offered a simulated GP map v 27 genotypic values we measured the degree of order-breaking for a certain locus *k* by the allele substitution effects at that locus. Because that a fixed background genotype at all various other loci (as shown in eq. (14) in <30>), us computed the difference in genotypic value once substituting a 0-allele through a 1-allele (i.e. When going native 00 come 01 or native 01 to 11 in ~ locus *k*). We built up substitution effects throughout all 9 elevator genotypes come compute *N*, the sum of all an unfavorable substitution effects, and *A*, the amount of absolute worths of every substitution effects. If the GP map is monotone because that locus *k* climate

*k*the .

### System classification and also phenotype dimensionality

The five cGP models studied in this work-related differ in numerous ways, both in their function and the basic network structure. The glycolysis and also cAMP models space both pathway models v an acyclic collection of reaction transforming inputs come outputs. The glycolysis version <20> is an ext advanced 보다 the metabolic models in previously genetically oriented research studies (e.g., <3>, <31>, <32>) together it contains many different types of enzyme kinetics too as an adverse feedback regulation of part enzyme tasks through product inhibition. The cAMP version <21> consists of a number of negative feedback loops, but overall it likewise has a clean pathway structure where the glucose signal is relayed from G-proteins come cAMP come the target kinase PKA. Both these 2 models have actually in common fairly simple dynamics with options converging to a stable steady state (Figure 1A and also B).

In contrast, the three other models present cyclic behavior resulting indigenous an interplay between positive and negative feedback loops (Figure 1 C–E). However, there are clear differences between these models too. The love cell model <24> is an excitable mechanism with feedback mechanisms consisting of calcium-induced calcium release and several voltage-dependent ion channels. In contrast to pacemaker cells, it relies on exterior pacing come initiate the activity potential. The circadian rhythm design <23>, <33> is a gene expression network with intertwined positive and an unfavorable transcriptional feedback loops, offering a border cycle oscillator with continual oscillations also in constant darkness. The cabinet cycle design <22> centers approximately a optimistic feedback loop in between B-type cyclins in association v cyclin dependent kinase and also inhibitors the the cyclin dependent kinase, which develops a hysteresis loop causing the cabinet cycle to emerge from transitions in between the two alternate stable stable states.

This crude classification of the 5 cGP models into pathway models and also more facility regulatory systems is clearly reflected in the reliable dimensionality the the phenotypes occurring in ours Monte Carlo simulations. Us studied the phenotypic dimensionality because that all five cGP models by primary Component analysis (PCA) because that each Monte Carlo simulation (Figure 2). Throughout all simulated datasets, 95% the phenotypic variation of the glycolysis and also cAMP models have the right to be defined by the an initial 3 principal components, the cabinet cycle and also heart cell models require the very first 5 primary components, and also 7 materials are required for the circadian model. Due to the fact that the genotype-to-parameter maps space additive because that all five models, these differences in the effective dimensionality that high-level phenotypes show that the mappings indigenous parameters to phenotypes are much easier for the pathway models 보다 the various other three models. This, in addition to results top top the impact of positive feedback on statistical epistasis in gene regulatory networks <11>, said that the glycolysis and also cAMP models might an outcome in greater *VA*/*VG* ratios 보다 the various other three models.

The proportion of full phenotypic variation defined (y axis) matches the variety of principal contents (x axis) throughout all 5 cGP models (colour coded). For each Monte Carlo data collection the

matrices comprise the complete genotype-phenotype map for all*M*tape-recorded phenotypes to be standardized come unit variance prior to principal materials analysis. Each boxplot summarizes defined variance because that close come 1000 Monte Carlo simulations.

https://doi.org/10.1371/journal.pcbi.1003053.g002

### The proportion of additive genetic variance to total genetic variance

The results evidenced our expectations regarding high *VA*/*VG* ratios because that the glycolysis and cAMP models. Furthermore, a number of distinct patterns emerged. The cAMP design shows the overall highest *VA*/*VG* ratios values (Figure 3A and also Table S6), with mean and also median values over 0.99 throughout all tape-recorded phenotypes. The 0.05-quantile (i.e. Just 5 percent the the Monte Carlo simulations show lower worths than this) *VA*/*VG* values were above 0.97 for all phenotypes and also no values lower than 0.6 to be observed. In various other words, an intra- and inter-locus additive design of gene action really well approximates the genotype-phenotype maps developing from this cGP model.

Figure 3. The empirical cumulative distribution function of

*VA*/

*VG*ratios for phenotypes the the cAMP (A) and also the glycolysis (B) models.

**A.** The empirical cumulative distribution functions (y axis) the *VA*/*VG* ratios (x axis) for every phenotypes studied in the cAMP model: The initial secure state concentration before adding external glucose the the cyclic adenosine monophosphate (cAMP), the G-protein Ras2a (Ras2a), the guanine-nucleotide-exchange element (Cdc25), the protein kinase A (PKAi). The height values after adding glucose of these proteins (cAMPv, Ras2av, Cdc25v and also PKAiv), the Kelch repeat homologue protein (Krhv), the G-protein Gpa2a (Gpa2av), and the phosphodiesterase (Pde1v). The time taken to reach the optimal values (cAMPt, Ras2at, Cdc25t, PKAit, Krht, Gpa2at, Ped1t). See Table S6 for further phenotype descriptions and also numerical summaries of the circulation of *VA*/*VG* ratios. **B.** The empirical cumulative distribution role (y axis) the *VA*/*VG* ratios (x axis) for the secure state concentration of 13 metabolites in the glycolysis model: acetaldehyde (ACE), 1,3-bisphospoglycerate (BPG), fructose-1,6-bisphosphate (F16P), furustos 6-phosphate (F6P), glucose 6-phosphate (G6P), glucose in cabinet (GLCi), nicotinamide adenine dinucleotide (NADH), phosphates in adenine nucleotide (P), 2-phosphoglyerate (P2G), 3-phosphoglycerate (P3G), phosphoenolpyruvate (PEP), pyruvate (PYP), and also trio-phosphate (TRIO). Watch Table S7 for more phenotype descriptions and also numerical summaries of the distribution of *VA*/*VG* ratios.

https://doi.org/10.1371/journal.pcbi.1003053.g003

The glycolysis model additionally has mean and median *VA*/*VG* worths close come 1 for all phenotypes (Figure 3B and Table S7). But compared come the cAMP model, the numbers are plainly lower; the shortest recorded typical value (phenotype *BPG*) is 0.9 and also 0.05-quantile worths are listed below 0.7 for some phenotypes. A few *VA*/*VG* values below 0.5 are observed for all phenotypes. The circulation of *VA*/*VG* ratios because that the cell cycle design (Figure S7 and also Table S8) is quite comparable to the of the glycolysis model, v a lowest median *VA*/*VG* worth of 0.93 for *time to peak* for *Sic1* and also with 0.05-quantiles below 0.8 for part phenotypes. *VA*/*VG* values listed below 0.1 room observed for a couple of Monte Carlo simulations in part phenotypes.

Figure 4. The empirical accumulation distribution function of

*VA*/

*VG*ratios because that phenotypes of the circadian version (A) and also the action potential model (B).

The empirical cumulative circulation functions (y axis) of *VA*/*VG* ratios (x axis) because that phenotypes learned in the circadian model and also the heart cabinet model. **A.** The upper-left dashboard (Bmal1) mirrors phenotypes concerned *bmal1* gene, including the mRNA (MB), the unphosphorylated/phosphorylated protein in cytosol (BC/BCP) and also nucleus (BN/BNP). The bottom-right dashboard (Per) is for *per* gene, including the mRNA (MP), the unphosphorylated protein (PC) and also the phosphorylated protein (PCP). The bottom concentration (solid line) and also the time take it to peak (dashed line) the each species are videotaped phenotypes. The bottom-left dashboard (Cry) is regarded *cry* gene, including the mRNA (MC), the unphosphorylated protein (CC) and phosphorylated protein (CCP). The upper-right dashboard (Complex) is because that protein complexes PCC, PCN, PCCP and PCNP. The period of circadian rate (Period, dotted line) is also shown. See Table S9 for more phenotype descriptions and numerical recaps of the distribution of *VA*/*VG* ratios. **B.** The empirical cumulative distribution functions (y axis) of *VA*/*VG* ratios (x axis) for phenotypes studied in the action potential model: time to 25%, 50%, 75% and 90% of initial values, the amplitude, initial worths (Base), peak values, time to reach peak values of activity potential (left panel) and calcium transient (right panel) room shown. Watch Table S10 for further phenotype descriptions and numerical summaries of the distribution of *VA*/*VG* ratios.

https://doi.org/10.1371/journal.pcbi.1003053.g004

All five cGP models are qualified of developing *VA*/*VG* ratios close come 1, and also except for two phenotypes because that the circadian model all average values the *VA*/*VG* are well over 0.5. This supports the hypothesis <30> that biological systems tend to involve regulation machinery that in general leads to significant additive genetic variance also at intermediate allele frequencies. The is not to to speak that choice cannot sometimes produce regulatory remedies that have tendency towards low *VA*/*VG* ratios; in fact, the incidence of short *VA*/*VG* ratios differed markedly amongst the 5 models that us studied. Since the genotype-parameter maps to be purely additive, all non-additive genetic variance is a result of non-linearity in the ODE models. The expected impact of introducing non-additivity in the genotype-parameter maps would be a additional decrease in the *VA*/*VG* ratios.

Our finding that models with facility regulation including positive feedback loops have tendency to give lower *VA*/*VG* agrees with a previous research on gene regulation networks <11>. Considering the fairly high *VA*/*VG* ratios of the cabinet cycle model compared to the circadian and action potential models, the complying with quote indigenous Tyson and also Novak"s <34> conversation of why the cell-cycle is much better understood together a hysteresis loop 보다 as a limit cycle oscillator (LCO), is extremely informative: “*Generally speaking, the duration of an LCO is a complicated function of every the kinetic parameters in the differential equations. However, the period of the cell department cycle counts on just one kinetic building of the cell: its mass-doubling time.*” This appears to define why the genotype-phenotype maps occurring from the cell-cycle models space much more linear 보다 the maps from the circadian model, i beg your pardon is one LCO.

### Monotonicity explains much the the VA/VG patterns

In a given population *VA*/*VG* is a duty of allele frequencies and also the GP map, and GP maps with strong interactions have the right to still give high *VA*/*VG* values in populations with too much allele frequencies <3>. In populaces with intermediary allele frequencies the *VA*/*VG* values are established mainly through the shape of the genotype-phenotype map, and also the it was observed differences in between the 5 cGP models in the circulation of *VA*/*VG* values urges a search for underlying explanatory principles.

The recently proposed ide of monotonicity (or order-preservation) the GP maps seems to be one together principle. In short, a GP map is claimed to be monotone if the notified of genotypes by gene content (the number of alleles of a given type) is kept in the notified of the linked phenotypic worths (see <30> because that details). Number 5 depicts three extreme varieties of GP maps checked out in our simulations. Almost additive GP maps as shown in figure 5A give *VA*/*VG* values an extremely close to one. GP maps with strong magnitude epistasis, however still order-preserving, typically result in intermediary *VA*/*VG* worths (Figure 5B), while very non-monotone or order-breaking GP maps (Figure 5C) showing solid overdominance and/or authorize epistasis result in *VA*/*VG* worths close come zero.

Examples of 3 distinct varieties of genotype-phenotype maps checked out in our simulations. For each subfigure the phenotypic value is presented on the y-axis when the x-axises, heat colours and also plot columns indicate the genotype in ~ the 3 loci. **A.** A almost additive map exemplified by the GP map of the time to peak concentration that Cdc25 (*VA*/*VG* = 0.99) in the cAMP model; **B.** A totally monotone yet non-additive map exemplified through the GP map of the concentration of P2G protein (*VA*/*VG* = 0.41) in the glycolysis model; and, **C.** A strongly non-monotonic map is found the time to height concentration that the pc protein (*VA*/*VG* = 0.03) from the circadian model.

https://doi.org/10.1371/journal.pcbi.1003053.g005

Based on recent results from studies of gene regulation networks <30>, we anticipated the the three cGP models with complicated regulation entailing positive feedback would an outcome in considerably much more non-monotone or order-breaking GP maps than the two pathway models. To check this, we measured the quantity of order-breaking in all simulated GP maps (see Methods) and found a an extremely clear pattern (Figure 6). When the datasets from the glycolysis and cAMP models had only 1.1% and also 1.3% GP maps with order-breaking for any kind of locus, those indigenous the cell cycle, circadian and activity potential models included 20.7%, 44.4% and 69.5%, respectively. Moreover, monotone GP maps gave greater *VA*/*VG* worths than non-monotone GP maps for all 5 cGP models (Mann-Whitney test; p-values listed below 1e-10 for all 5 models).

The number of Monte Carlo simulations whereby the GP-map for a given phenotype is plainly order-breaking (GP maps v *N/A*>0.05, watch Methods) is presented for the cAMP design (**A**), the glycolysis model (**B**), the cell cycle design (**C**), the circadian design (**D**) and also the action potential model (**E**). Just phenotypes v at least one Monte carlo simulation leading to an order-breaking GP map space shown.

https://doi.org/10.1371/journal.pcbi.1003053.g006

However, regardless of the reality that the glycolysis design rarely reflects order-breaking also for a single locus, it own much reduced *VA*/*VG* worths than the cAMP model. A plausible explanation is that the steady-state concentrations of metabolites have the right to markedly boost for parameter values close come a saddle-node bifurcation allude <26>. Simulation outcomes v unstable steady states were discarded, but in those situations where one of the genotypes (i.e. Parameter sets) come close come the bifurcation allude without crossing it we gain genotype-phenotype maps as in number 5B, whereby one genotype (or a tiny set) provides much higher phenotypic values than the others. Such GP maps, similar to the duplicate aspect model in Hill *et al.* <3>, are known to provide low *VA*/*VG* ratios regardless of being monotonic. Comparable GP maps giving *VA*/*VG* ratios close to zero were likewise found by Keightley <32> in his research of metabolic models own null alleles at all loci.

### Considerations top top the genericity of the results

Our main reason for restricting the sampled hereditary variation that parameters to within 30% that the published baseline values was to avoid qualitative (or topological) transforms of the dynamics. Together qualitative transforms are regularly biologically realistic descriptions of knockouts or other huge genetic changes, because that example activity potentials of alternative amplitude (alternans) <17>; ns of stable circadian oscillation <23>; and non-viable cell-cycle mutants phenotypes <22>. However, since the heritability and also variance component concepts are identified for phenotypes showing consistent rather 보다 discrete variation, we sought to prevent such qualitative alters here.

We ran simulations with five polymorphic loci for the cAMP (Figure S8A), glycolysis (Figure S8B), cell cycle (Figure S9) and action potential (Figure S10) models (the circadian model defines only three genes explicitly). The resulting *VA*/*VG* worths were slightly lower than with 3 loci, however the in its entirety shape the the distributions and the clear differences between models did no change. This indicates that ours findings room of basic relevance because that oligogenic traits.

It have to be emphasized that the 5 studied cGP models differ in several other facets than those highlighted here, such together the device size (number the state variables) and also the process time scales. These features could likewise contribute to the it was observed variation in the distributions of *VA*/*VG* ratios. However, together structural distinctions are inescapable when the target is to to compare experimentally validated models designed come describe certain biological systems. A complementary approach is to study generic models where device size and equation framework is fixed, while the connectivity matrix have the right to be changed to explain a family of systems <35>. This facilitates graph-theoretic comparison of systems at the expense of some organic realism. Us anticipate that the major conclusions indigenous such researches will be similar to ours, however it may an extremely well be that other crucial generic insights may additionally come to the fore.

All the models in our study explain parts of the to move machinery and the result phenotypes are hence cellular rather than organismal. We perform not think this is a major shortcoming in regards to the key conclusions that arise from our results. However, we anticipate that applications of our strategy on multiscale models consisting of cellular, tissue and also whole-organ phenotypes <36> will administer a much improved structure for explaining just how properties the the GP map vary throughout and within organic systems in regards to regulatory anatomy and also associated genetic variation <37>, <38>.

As our technique can it is in used together with any computational biological model, it has actually the potential to add substantially come a theoretical foundation capable of predicting when we are to suppose low or high *VA*/*VG* or *h2*/*H2* ratios indigenous the principles of regulatory biology. Causally cohesive genotype-phenotype modeling thus shows up to qualify as a promising strategy for completely causal models of biological networks and also physiology with quantitative genetics <39>–<44>.

### Figure S1.

**Flowchart that Monte Carlo simulations and also analysis.** Flowchart the the Monte Carlo simulations explained in the techniques section “Monte Carlo simulations” and also subsequent evaluation described in the techniques section “Statistical analysis”.

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### Figure S2.

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### Figure S3.

**Graphical depiction of glycolysis model.** figure modified indigenous the CellML design repository (http://models.cellml.org/workspace/teusink_passarge_reijenga_esgalhado_vanderweijden_schepper_walsh_bakker_vandam_westerhoff_snoep_2000). Red numbers, correspond to the rows in Table S2, and also indicate the model aspects where genetic variation to be introduced.

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### Figure S4.

**Graphical representation of cell cycle model.** figure modified native the CellML design repository (http://models.cellml.org/workspace/chen_calzone_csikasznagy_cross_novak_tyson_2004). Red numbers, correspond to the rows in Table S3, and also indicate the model facets where genetic variation to be introduced.

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### Figure S5.

**Graphical representation of circadian model.** figure modified from the CellML version repository (http://models.cellml.org/workspace/leloup_goldbeter_2004). Red numbers, exchange mail to the rows in Table S4, and also indicate the model aspects where genetic variation to be introduced.

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### Figure S6.

**Graphical depiction of activity potential model.** number modified from the CellML design repository (http://models.cellml.org/workspace/bondarenko_szigeti_bett_kim_rasmusson_2004). Red numbers, exchange mail to the rows in Table S5, and also indicate the model aspects where genetic variation was introduced.

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### Figure S7.

**The empirical accumulation distribution role of ***VA***/***VG*** ratios because that phenotypes that the cell cycle model.** The empirical cumulative distribution functions (y axis) that *VA*/*VG* ratios (x axis) for all phenotypes learned in the cabinet cycle model. The phenotypes are divided into 3 groups. Cell events refer come the discrete events identified in the model document and incorporate timing the budding (Bud), time of DNA replication (Rep) and timing that alignment of chromosomes on the metaphase bowl (Spn). Optimal concentration include the concentration the the phosphorylated anaphase-promoting complicated (APCP), the G1-stabilizing protein Cdc6, the B-type Cyclin protein 2 (Clb2), the S-phase fostering B-type Cyclin (Clb5), the starter kinase (Cln2) and the G1 phase stabilizing protein (Sci1). The moment to optimal phenotypes include the time come reach peak concentrations that APCP, Cdc6, Clb2, Clb5, Cln2 and Sci1. Check out Table S8 for more phenotype descriptions and numerical recaps of the distribution of *VA*/*VG* ratios.

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### Figure S8.

**The empirical accumulation distribution role of ***VA***/***VG*** ratios for phenotypes the the cAMP (A) and also the glycolysis (B) models v 5 polymorphic loci.** number 3 mirrors results indigenous simulations v 3 polymorhpic loci. **A.** The empirical cumulative circulation functions (y axis) of *VA*/*VG* ratios (x axis) for every phenotypes studied in the cAMP model: The initial stable state concentration before including external glucose of the cyclic adenosine monophosphate (cAMP), the G-protein Ras2a (Ras2a), the guanine-nucleotide-exchange aspect (Cdc25), the protein kinase A (PKAi). The peak values after including glucose of these proteins (cAMPv, Ras2av, Cdc25v and PKAiv), the Kelch repeat homologue protein (Krhv), the G-protein Gpa2a (Gpa2av), and the phosphodiesterase (Pde1v). The time required to reach the optimal values (cAMPt, Ras2at, Cdc25t, PKAit, Krht, Gpa2at, Ped1t). **B.** The empirical cumulative distribution function (y axis) of *VA*/*VG* ratios (x axis) because that the steady state concentration of 13 metabolites in the glycolysis design acetaldehyde (ACE), 1,3-bisphospoglycerate (BPG), fructose-1,6-bisphosphate (F16P), fructose 6-phosphate (F6P), glucose 6-phosphate (G6P), glucose in cell (GLCi), nicotinamide adenine di nucleotide (NADH), phosphates in adenine nucleotide (P), 2-phosphoglyerate (P2G), 3-phosphoglycerate (P3G), phosphoenolpyruvate (PEP), pyruvate (PYP), and also trio-phosphate (TRIO).

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### Figure S9.

**The empirical accumulation distribution duty of ***VA***/***VG*** ratios for phenotypes the the cell cycle design with 5 polymorphic loci.** figure S7 shows results indigenous simulations with 3 polymorhpic loci. The empirical cumulative distribution functions (y axis) the *VA*/*VG* ratios (x axis) for every phenotypes learned in the cell cycle model. The phenotypes are divided into 3 groups. Cell occasions refer to the discrete events defined in the model file and encompass timing that budding (Bud), timing of DNA replication (Rep) and also timing the alignment of chromosomes on the metaphase plates (Spn). Optimal concentration include the concentration of the phosphorylated anaphase-promoting complex (APCP), the G1-stabilizing protein Cdc6, the B-type Cyclin protein 2 (Clb2), the S-phase promoting B-type Cyclin (Clb5), the starter kinase (Cln2) and also the G1 phase stabilizing protein (Sci1). The time to optimal phenotypes encompass the time to reach height concentrations of APCP, Cdc6, Clb2, Clb5, Cln2 and Sci1.

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### Figure S10.

**The empirical cumulative distribution duty of ***VA***/***VG*** ratios because that phenotypes that the action potential version with 5 polymorphic loci.** number 4B shows results native simulations v 3 polymorhpic loci. The empirical cumulative distribution functions (y axis) the *VA*/*VG* ratios (x axis) because that phenotypes studied in the action potential model: time to 25%, 50%, 75% and also 90% of early values, the amplitude, initial worths (Base), optimal values, time come reach peak values of activity potential (left panel) and calcium transient (right panel) space shown.

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### Table S1.

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### Table S2.

**Polymorphic model aspects of the glycolysis model.** A list of glycolysis model elements and parameters used to manifest hereditary variation. Parameter names native the initial publication <20>, names offered in the SBML paper retrieved from http://www.ebi.ac.uk/biomodels-main/BIOMD0000000064 and baseline values through units.

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### Table S3.

**Polymorphic model facets of the cell cycle model.** A perform of cabinet cycle model elements and also parameters supplied to manifest hereditary variation. Parameter names indigenous Table 1 and Table 2 in the initial publication <22>, names supplied in the CellML file retrieved indigenous http://models.cellml.org/workspace/chen_calzone_csikasznagy_cross_novak_tyson_2004 and also baseline values v units.

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### Table S4.

**Polymorphic model aspects of the circadian model.** A perform of circadian design elements and also parameters used to manifest hereditary variation. Parameter names from Table 1 (parameter set 4) in the initial publication <23>, names used in the CellML document “leloup_goldbeter_2004.cellml” retrieved native http://models.cellml.org/workspace/leloup_goldbeter_2004/ and baseline values with units.

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### Table S5.

**Polymorphic model aspects of the action potential model.** A perform of action potential model elements and also parameters supplied to manifest hereditary variation. Parameter names from Table B1 in the original publication <24>, names supplied in the CellML document which is accessible as supplementary material (filename “LNCS model.zip”) at doi:10.3389/fphys.2011.00106 and baseline values v units.

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### Table S6.

**Summary that phenotype descriptions, variability thresholds and also distribution of ***VA***/***VG*** ratios for the cAMP model.** The very first three columns perform the phenotype abbreviations used in this study, a text summary of the phenotypes and also their units. The thresholds supplied to filter the end dataset with an extremely low loved one and/or absolute variability are detailed in the following two columns, adhered to by the number of Monte Carlo simulations (out of 1000) passing the threshold. The critical 7 columns contain quantiles and method of the *VA*/*VG* values for the datasets pass the variability threshold.

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### Table S7.

**Summary the phenotype descriptions, variability thresholds and distribution that ***VA***/***VG*** ratios for the glycolysis model.** The first three columns list the phenotype abbreviations offered in this study, a text summary of the phenotypes and their units. The thresholds provided to filter the end dataset with very low family member and/or absolute variability are detailed in the following two columns, complied with by the number of Monte Carlo simulations (out of 1000) happen the threshold. The last 7 columns save on computer quantiles and method of the *VA*/*VG* values for the datasets passing the variability threshold.

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### Table S8.

**Summary that phenotype descriptions, variability thresholds and distribution the ***VA***/***VG*** ratios for the cell cycle model.** The first three columns list the phenotype abbreviations provided in this study, a text description of the phenotypes and their units. The thresholds offered to filter the end dataset with very low family member and/or absolute variability are listed in the next two columns, followed by the number of Monte Carlo simulations (out the 1000) happen the threshold. The last 7 columns contain quantiles and method of the *VA*/*VG* worths for the datasets passing the variability threshold.

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### Table S9.

**Summary the phenotype descriptions, variability thresholds and distribution the ***VA***/***VG*** ratios for the circadian model.** The first three columns list the phenotype abbreviations provided in this study, a text summary of the phenotypes and their units. The thresholds used to filter the end dataset with an extremely low relative and/or absolute variability are detailed in the next two columns, complied with by the variety of Monte Carlo simulations (out that 1000) pass the threshold. The critical 7 columns save quantiles and way of the *VA*/*VG* worths for the datasets passing the variability threshold. Abbreviations: phosphorylated – phos., cytosolic – cyt., atom – nuc., bottom concentration – b.c., top concentration – p.c.

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### Table S10.

**Summary the phenotype descriptions, variability thresholds and distribution that ***VA***/***VG*** ratios because that the activity potential model.** The first three columns perform the phenotype abbreviations provided in this study, a text summary of the phenotypes and also their units. The thresholds provided to filter out dataset with very low family member and/or pure variability are noted in the next two columns, followed by the number of Monte Carlo simulations (out of 1000) happen the threshold. The last 7 columns contain quantiles and way of the *VA*/*VG* values for the datasets pass the variability threshold.

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### Text S1.

**More comprehensive descriptions that the 5 cGP models.**

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## Acknowledgments

We are thankful to Katherine C. Chen for assist on implementing and also solving the cabinet cycle model.

See more: What Channel Is Fx On Dish Tv Guide & Channel Lineup, Fx On Dish Network

## Author Contributions

Conceived the study: ABG SWO. Carry out simulations and also analysis: YW ABG JOV. Wrote the paper: YW JOV SWO ABG.

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