impact of Regulatory architecture on broad versus Narrow sense Heritability Yunpeng Wang, Jon Olav Vik, Stig W. Omholt, Arne B. Gjuvsland
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Additive genetic variance (VA) and also total genetic variance (VG) room core ideas in biomedical, evolutionary and production-biology genetics. What determines the large variation in reported VA/VG ratios indigenous line-cross experiments is no well understood. Below we report exactly how the VA/VG ratio, and also thus the ratio in between narrow and large sense heritability (h2/H2), varies together a role of the regulatory style underlying genotype-to-phenotype (GP) maps. Us studied five dynamic models (of the cAMP pathway, the glycolysis, the circadian rhythms, the cabinet cycle, and heart cabinet dynamics). We assumed genetic variation to be reflect in version parameters and also extracted phenotypes summarizing the device dynamics. Even when imposing purely direct genotype come parameter maps and no ecological variation, us observed rather low VA/VG ratios. In particular, equipment with confident feedback and cyclic dynamics gave more non-monotone genotype-phenotype maps and much lower VA/VG ratios 보다 those without. The results show that part regulatory architectures consistently maintain a transparent genotype-to-phenotype relationship, whereas various other architectures generate much more subtle patterns. Our strategy can be provided to elucidate these relationships across a whole selection of biological systems in a systematic fashion.

You are watching: Broad sense vs narrow sense heritability


The broad-sense heritability that a characteristics is the ratio of phenotypic variance attributable to genetic causes, while the narrow-sense heritability is the ratio attributable to additive gene effects. A far better understanding of what underlies variation in the proportion of the 2 heritability measures, or the equivalent ratio of additive variance VA to total genetic variance VG, is necessary for production biology, biomedicine and evolution. We uncover that report VA/VG worths from heat crosses differ greatly and ask if biological mechanisms basic such differences can be elucidated by linking computational biology models through genetics. Come this end, we made use of models the the cAMP pathway, the glycolysis, circadian rhythms, the cell cycle and cardiocyte dynamics. We assumed additive gene action from genotypes to design parameters and also studied the resulting GP maps and VA/VG ratios of system-level phenotypes. Our results show that some varieties of regulation architectures consistently maintain a transparent genotype-to-phenotype relationship, whereas rather generate much more subtle patterns. Particularly, equipment with confident feedback and cyclic dynamics result in an ext non-monotonicity in the GP map leading to lower VA/VG ratios. Our technique can be supplied to elucidate the VA/VG relationship across a whole range of biological systems in a methodical fashion.


Citation: Wang Y, Vik JO, Omholt SW, Gjuvsland ab (2013) impact of Regulatory style on wide versus Narrow sense Heritability. rebab.net Comput Biol 9(5): e1003053. Https://doi.org/10.1371/journal.pcbi.1003053

Editor: William Stafford Noble, university of Washington, United says of America

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

Funding: This work was supported by the research study Council that Norway (http://www.rcn.no) under the eVITA program, task number 178901/V30, and also by the virtual Physiological Rat job (http://virtualrat.org) funded with NIH grant P50-GM094503. The development of the cgptoolbox was supported by the VPH-Network the Excellence (http://vph-noe.eu) with exemplar job EP7. NOTUR, the Norwegian metacenter for computational science, provided computing resources under task nn4653k. The funders had no duty in research design, data collection and also analysis, decision to publish, or preparation of the manuscript.

Competing interests: The writer have asserted that no completing interests exist.


Introduction

The broad-sense heritability that a trait,

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, is the ratio of phenotypic variance attributable to hereditary causes, while the narrow-sense heritability
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, is the relationship attributable to additive gene action. The complete genetic variance
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consists of the variance defined by intra-locus dominance (
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) and inter-locus interaction (
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). The reasons for and also importance that this non-additive genetic variance the distinguishes the two heritability measures has actually been topic to considerable controversy for an ext than 80 year (e.g., <1>–<6>). It was recently presented through statistical disagreements that for traits with countless loci at too much allele frequencies, much of the genetic variance becomes additive v h2/H2 (or equivalently VA/VG) frequently >0.5 <3>. In populations with intermediary allele frequencies, such as managed line crosses, reduced VA/VG ratios are often reported <7>, <8>. This is depicted in Table 1, which summarizes estimated VA/VG ratios from a repertoire of studies on such populations. This wide selection of h2/H2 ratios reported for line crosses cannot be explained by one allele-frequency argument, and also putative explanations must be based on how the regulatory style of the underlying biological systems form the genotype-phenotype (GP) map.


Download:
Table 1. examples of reported VA/VG ratios of indigenous line-crossing experiments.

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


It is important to know the causal underpinnings of the observed variation in h2/H2 ratios within and between organic systems for several reasons. In human being quantitative genetics, whereby twin researches are commonly used, many heritability estimates refer to H2 <9>. In cases where h2/H2 is low, this have the right to lead come unrealistic expectations around how lot of the underlying causative variation may be situated by linear QTL detection methods <6>. Top top the various other hand, short narrow feeling heritability because that a given facility trait does not necessarily suggest that the environment determines much of the variation. In evolutionary biology, additive variance is the foremost currency for evolution adaptation and also evolvability. Important questions in this context room for instance (i) to which level is there an option on the regulation anatomies us to maintain high additive variance, (ii) are there organizational constraints in structure adaptive systems such that in some cases a low h2/H2 ratio have to of necessity arise while the proximal systems is tho selected for? Moreover, in manufacturing biology v genetically modified, sexually reproducing organisms, one would like to ensure that the adjustments would it is in passed end to future generations in a totally predictable way. Thus, one would favor to ensure the the alteration becomes very heritable in the small sense.

As a step towards a physiologically grounded expertise of the sport of the h2/H2 relationship throughout biological equipment or processes, us posed the question: room there regulatory structures, or particular classes that phenotypes, an ext likely to generate low VA/VG ratios 보다 others? Addressing this concern requires the linking of hereditary variation to computational biological in a populace context (e.g., <10>–<19>), so-called causally-cohesive genotype-phenotype (cGP) modeling <15>, <17>, <18>. We applied this method to five well-validated computational biological models describing, respectively, the glycolysis metabolic pathway in budding yeast <20>, the cyclic adenosine monophosphate (cAMP) signaling pathway in budding yeast <21>, the cabinet cycle regulation the budding yeast <22>, the gene network underlying mammalian circadian rhythms <23>, and the ion channels determining the activity potential in computer mouse heart myocytes <24> these models different in your regulatory architecture; below, we display that they additionally differ in the selection of VA/VG ratios that they can exhibit. In particular, confident feedback regulation and oscillatory behavior seem to dispose for short VA/VG ratios. The results indicate that our approach can be used in a generic manner come probe exactly how the h2/H2 ratio varies as a role of regulation anatomy.


Simulations the cGP models

The 5 cGP models were built and also analyzed through the cgptoolbox (http://github.com/jonovik/cgptoolbox) an open-source Python package occurred by the authors; further resource code particular to the simulations in this record is available on request. In the following we explain the three main parts that the workflow: (i) the mapping indigenous genotypes come parameters, (ii) the mapping native parameters come phenotypes, i.e. Solving the dynamic models and (iii) the setup the Monte-Carlo simulations combine the 2 mappings (Figure S1). Because that each model, we briefly explain its origins, the software offered to settle 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 mirrors graphical representations of the 5 model systems and also Text S1 contains an ext detailed explanation of all 5 models.


Genotype to parameter mapping.

For each model, the adhering to procedure was recurring 1000 time (see also “Monte Carlo simulations” below) for different selections that parameters come be based on simulated genetic variation. We started by sampling 3 polymorphic loci, each determining one or two parameters in the dynamic model. Tables of eligible loci with equivalent parameters and also their baseline worths are listed in Table S1, S2, S3, S4, S5, matching to the cAMP, glycolysis, cabinet cycle, circadian and activity potential models respectively. Heritable sports in a preferred parameter was created for a solitary biallelic locus through allele indexes 0 and also 1 in the adhering to manner. First, two numbers r1 and also r2 to be sampled uniformly in the expression <0.7, 1.3>. The parameter worth for a homozygote 00 was set to

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where b is the baseline value, for a homozygote 11 the parameter worth was
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. The heterozygous genotype 01 was assigned the mean of the two homozygotes
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, resulting in an additive mapping from genotypes come parameter values.


cAMP model.

Graphical illustration of the phenotypes videotaped for the 5 cGP models studied. Time process (state change on y-axis, time top top x-axis) because that the baseline parameter set are presented for all 5 models. A. In the absence of external glucose all state variables (concentration the cAMP is shown) in the cAMP version <21> converge come a stable steady state (blue circle on y-axis). After addition of outside glucose (5 mM added at time 50) we see a rapid adjust followed by a slow return to the original steady state. In enhancement to the original steady state, the extremal concentration (top blue circle) as well as the time to with the extremum (blue line) was tape-recorded as phenotypes. B. Metabolite concentrations (internal glucose (GLCi), glucose-6-phospate (G6P) and also fructose-6-phospate (F6P) room shown) in the glycolysis version <20> every converge come a steady steady state, suggested by open circles. The steady state concentrations because that 13 metabolites were videotaped as phenotypes indigenous this model. C. because that the cell cycle model <22> we tape-recorded the height level and the time indigenous bottom to peak as because that the circadian version (Figure 1D), and also in addition we taped cell cycle events such as bud development at the time as soon as  = 1 indicated by the black color arrow. D. mRNA and also protein concentration (mRNA because that Bmal1 (MB), Cry (MC) and Per (MP) space shown) in the circadian design <23> converge to a border cycle. In enhancement to the duration of oscillation (long blue line) because that each the the 16 variables the peak level (open blue circle) as well as the time native bottom to top (short blue line) were videotaped as phenotypes. E. We offered the basic level, height level, amplitude, time to peak, and time to 25%, 50%, 75% and also 90% restore of the activity potential and calcium transient together cell level phenotypes that the activity potential model <24>. An action potential is displayed in the figure.

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Glycolysis model.

The design published by Teusink et al. <20> explains glycolysis in S. Cerevisiae with the kinetics the 13 glycolytic enzymes determining the fluxes that metabolite state variables. Hereditary variation was introduced on maximal reaction prices for the enzymes (see figure S3 and Table S2). Us downloaded the model from the BioModels database (http://www.ebi.ac.uk/biomodels-main/BIOMD0000000064) in SBML L2 V1, and solved it with PySCeS <25> to uncover the stable steady state concentrations of metabolites, i m sorry were offered as phenotypes (see number 1B and Table S7). Datasets to be discarded if one or an ext of the genotypes walk not give a steady steady state, as can happen due to a saddle-node bifurcation <26>.


Cell bicycle model.

The version of the agreement control instrument of the cabinet cycle in S. Cerevisae modeled by algebraic/differential equations that explain the consistent changes in state variables and discontinuities because of cellular occasions <22> was acquired from the CellML repository (http://models.cellml.org/workspace/chen_calzone_csikasznagy_cross_novak_tyson_2004). Genetic variation was introduced on the production and also decay prices of various proteins (see figure S4 and also Table S3). The published model includes reset rules (events) because that both parameters and also state variables, yet the CellML implementation only includes the parameter (kmad2, kbub2 and klte1) rules. Reset rules because that state variables , , and also as explained in the design paper, were enforced by fixing the version with rootfinding for the pertinent variables. The design was numerically combined using the CVODE solver <27> till convergence of cell division time, cabinet cycle events. The top levels and also time to height levels for the cytosolic protein concentrations, in addition to the timing of cell department events were taped as phenotypes (see figure 1C for phenotype illustrations and Table S8 for phenotype descriptions).


Circadian model.

The model of the mammalian circadian clock released by Leloup and Goldbeter <23> explains the dynamics the mRNA and proteins of three genes in the cytosol and also nucleus. Genetic variation was introduced on mRNA degeneration rates (see number S5 and also Table S4). The design was download from CellML repository (http://models.cellml.org/workspace/leloup_goldbeter_2004) and integrated utilizing CVODE <27> till convergence come its limit cycle. As phenotypes we offered the bottom levels and time to from bottom level to peak value of the concentration of mRNAs, proteins and protein complexes. In addition, we videotaped the period of oscillations (see number 1D because that phenotype illustrations and Table S9 for phenotype descriptions).


Action potential model.

The version of <24> is an expansion of <28> and also describes the action potential and also calcium transient the a computer mouse heart muscle cell. We obtained CellML password from the authors and also the paper is available as supplementary material in <17>. Number integration was done using CVODE <27>. Hereditary variation was presented on the maximal conductances that ion channels and pump affinities (see figure S6 and Table S5). Phenotypes were created by simulated consistent pacing together done in <17>, <18>, through a economic stimulation potassium current of −15 V/s was lasting for 3 ms in ~ the begin of every stimulus interval. The model was simulated to convergence as explained in <17>; datasets that failed to converge were discarded. The early stage level, optimal level, amplitude, and also time come 25, 50, 75 and 90% recovery of the activity potential and calcium transient were taped as the cabinet level phenotypes (see number 1E for phenotype illustrations and also Table S10 for phenotype descriptions).


Monte Carlo simulations.

For each model we performed 1000 Monte Carlo simulations as follows (see figure S1 for an illustration). We an initial sampled three polymorphic loci for arrival of hereditary variation and also sampled the genotype-to-parameter map as explained above. Then all 27 possible three-locus genotypes were enumerated, mapped into 27 parameter sets and also for every parameter collection the dynamic design was solved and also phenotypes (as described over and in number 1) were obtained. To stop artifacts arising from numerical noise datasets through low variability to be omitted from the hereditary analysis. Absolute variability was measured together the difference in between the maximum and also minimum worths of a phenotype in a dataset, and relative variability as the ratio of the absolute variation to the average phenotype that the very same dataset. The threshold values for each phenotype and also the variety of datasets exceeding the thresholds are provided in Tables S6, S7, S8, S9, S10, because that the cAMP, glycolysis, cabinet cycle, circadian and activity potential models, respectively.


Decomposition of genetic variance.

A single Monte Carlo simulation results in genotype-to-phenotype maps made up by 27 genotypic values (i.e. The phenotype values corresponding to the 27 genotypes) because that a given phenotype. We used the NOIA framework <29> come compute the resulting hereditary variance (

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) in a hypothetical F2 populace and decompose it right into additive (
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) and non-additive components (
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and also
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). This was done v the role linearGPmapanalysis in the R package noia (http://cran.r-project.org/web/packages/noia/) variation 0.94.1.


Monotonicity that GP-maps.

We develop on the meanings of monotonicity and also the indexing that alleles presented in <30>. Offered a simulated GP map with 27 genotypic worths we measure the degree of order-breaking because that a certain locus k through the allele substitution effects at that locus. Because that a resolved background genotype at all various other loci (as suggested in eq. (14) in <30>), us computed the difference in genotypic value when substituting a 0-allele with a 1-allele (i.e. When going from 00 come 01 or native 01 come 11 in ~ locus k). We built up substitution effects across all 9 background genotypes to compute N, the amount of all an unfavorable substitution effects, and A, the amount of absolute values of all substitution effects. If the GP map is monotone because that locus k then

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, and also if it is order-breaking because that locus k the
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.


System classification and also phenotype dimensionality

The 5 cGP models learned in this job-related differ in numerous ways, both in their role and the underlying network structure. The glycolysis and cAMP models space both pathway models v an acyclic series of reaction transforming inputs to outputs. The glycolysis version <20> is more advanced 보다 the metabolic models in previously genetically oriented research studies (e.g., <3>, <31>, <32>) together it has many different varieties of enzyme kinetics also as an adverse feedback regulation of some enzyme activities through product inhibition. The cAMP design <21> includes a number of an unfavorable feedback loops, but overall it also has a clean pathway framework where the glucose signal is relayed native G-proteins to cAMP come the target kinase PKA. Both these 2 models have actually in common fairly simple dynamics with solutions converging to a secure steady state (Figure 1A and also B).

In contrast, the three other models display cyclic actions resulting native an interplay between positive and an unfavorable feedback loops (Figure 1 C–E). However, there are clear differences between these models too. The love cell model <24> is one excitable device with feedback mechanisms including calcium-induced calcium release and several voltage-dependent ion channels. In comparison to pacemaker cells, it counts on external pacing come initiate the activity potential. The circadian rhythm version <23>, <33> is a gene expression network with linked positive and negative transcriptional feedback loops, providing a border cycle oscillator with continual oscillations also in constant darkness. The cabinet cycle version <22> centers about a confident feedback loop between B-type cyclins in association with cyclin dependency kinase and inhibitors of the cyclin dependent kinase, which develops a hysteresis loop resulting in the cell cycle to emerge from transitions in between the two different stable stable states.

This crude group of the five cGP models right into pathway models and more complicated regulatory equipment is clearly reflected in the effective dimensionality that the phenotypes arising in ours Monte Carlo simulations. Us studied the phenotypic dimensionality because that all five cGP models by major Component evaluation (PCA) because that each Monte Carlo simulation (Figure 2). Throughout all simulated datasets, 95% that phenotypic variation of the glycolysis and cAMP models deserve to be explained by the an initial 3 primary components, the cell cycle and heart cabinet models call for the very first 5 principal components, and also 7 materials are required for the circadian model. Because the genotype-to-parameter maps space additive because that all 5 models, these differences in the effective dimensionality of high-level phenotypes show that the mappings native parameters to phenotypes are easier for the pathway models 보다 the other three models. This, in addition to results on the result of confident feedback on statistics epistasis in gene regulatory networks <11>, suggested that the glycolysis and also cAMP models might result in higher VA/VG ratios 보다 the other three models.


The proportion of total phenotypic variation defined (y axis) versus the variety of principal materials (x axis) across all five cGP models (colour coded). Because that each Monte Carlo data set the

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matrices containing the full genotype-phenotype map for every M videotaped phenotypes was standardized to unit variance prior to principal contents analysis. Each boxplot summarizes defined variance because that close come 1000 Monte Carlo simulations.

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The ratio of additive hereditary variance to total genetic variance

The results evidenced our expectations concerning high VA/VG ratios for the glycolysis and cAMP models. Furthermore, a variety of distinct trends emerged. The cAMP version shows the overall highest VA/VG ratios worths (Figure 3A and also Table S6), v mean and also median values above 0.99 throughout all recorded phenotypes. The 0.05-quantile (i.e. Only 5 percent of the Monte Carlo simulations display lower worths than this) VA/VG values were above 0.97 for every phenotypes and no values lower than 0.6 were observed. In various other words, one intra- and also inter-locus additive model of gene action very well approximates the genotype-phenotype maps emerging 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 circulation functions (y axis) of VA/VG ratios (x axis) for every phenotypes learned in the cAMP model: The initial steady state concentrations before adding external glucose of the cyclic adenosene monophosphate (cAMP), the G-protein Ras2a (Ras2a), the guanine-nucleotide-exchange variable (Cdc25), the protein kinase A (PKAi). The optimal values after adding glucose of this 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 peak values (cAMPt, Ras2at, Cdc25t, PKAit, Krht, Gpa2at, Ped1t). View Table S6 for more phenotype descriptions and numerical recaps of the distribution of VA/VG ratios. B. The empirical cumulative distribution function (y axis) of VA/VG ratios (x axis) for the steady 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 cell (GLCi), nicotinamide adenine di nucleotide (NADH), phosphates in adenin nucleotide (P), 2-phosphoglyerate (P2G), 3-phosphoglycerate (P3G), phosphoenolpyruvate (PEP), pyruvate (PYP), and also trio-phosphate (TRIO). See Table S7 for more phenotype descriptions and numerical recaps of the distribution of VA/VG ratios.

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


Figure 4. The empirical cumulative distribution function of VA/VG ratios for phenotypes that the circadian version (A) and also the action potential design (B).

The empirical cumulative distribution functions (y axis) that 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 related to bmal1 gene, including the mRNA (MB), the unphosphorylated/phosphorylated protein in cytosol (BC/BCP) and nucleus (BN/BNP). The bottom-right panel (Per) is because that 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 top (dashed line) that each species are taped phenotypes. The bottom-left panel (Cry) is related to cry gene, including the mRNA (MC), the unphosphorylated protein (CC) and phosphorylated protein (CCP). The upper-right dashboard (Complex) is for protein complexes PCC, PCN, PCCP and PCNP. The period of circadian valuation (Period, dotted line) is likewise shown. View Table S9 for further phenotype descriptions and also numerical recaps of the distribution of VA/VG ratios. B. The empirical cumulative distribution functions (y axis) that VA/VG ratios (x axis) for phenotypes studied in the activity potential model: time to 25%, 50%, 75% and 90% of early values, the amplitude, initial worths (Base), optimal values, time to reach height values of action potential (left panel) and calcium transient (right panel) room shown. Check out Table S10 for further phenotype descriptions and numerical summaries of the circulation of VA/VG ratios.

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All five cGP models are qualified of creating VA/VG ratios close to 1, and except for 2 phenotypes because that the circadian version all average values of VA/VG space well over 0.5. This support the hypothesis <30> that biological systems have tendency to involve regulatory machinery that in general leads to considerable additive genetic variance also at intermediary allele frequencies. That is not to speak that an option cannot sometimes develop regulatory solutions that often tend towards short VA/VG ratios; in fact, the incidence of short VA/VG ratios differed markedly among the 5 models that we studied. Because the genotype-parameter maps to be purely additive, every non-additive hereditary variance is a an outcome of non-linearity in the ODE models. The expected result of introducing non-additivity in the genotype-parameter maps would certainly be a more decrease in the VA/VG ratios.

Our finding that models with complex regulation including positive feedback loops often tend to offer lower VA/VG agrees through a previous examine on gene regulatory networks <11>. Considering the relatively high VA/VG ratios of the cabinet cycle model contrasted to the circadian and activity potential models, the complying with quote native Tyson and also Novak"s <34> discussion of why the cell-cycle is better understood together a hysteresis loop than as a limit cycle oscillator (LCO), is very informative: “Generally speaking, the duration of an LCO is a complicated function of all the kinetic parameters in the differential equations. However, the duration of the cell department cycle relies on just one kinetic building of the cell: that is mass-doubling time.” This appears to explain why the genotype-phenotype maps occurring from the cell-cycle models are much much more linear 보다 the maps native the circadian model, which is an LCO.


Monotonicity defines much the the VA/VG patterns

In a given populace VA/VG is a function of allele frequencies and the GP map, and also GP maps with strong interactions can still offer high VA/VG values in populations with extreme allele frequencies <3>. In populaces with intermediate allele frequencies the VA/VG values are established mainly by the shape of the genotype-phenotype map, and also the observed differences between the 5 cGP models in the circulation of VA/VG values motivates a find for basic explanatory principles.

The recently proposed ide of monotonicity (or order-preservation) the GP maps appears to it is in one together principle. In short, a GP map is claimed to it is in monotone if the bespeak of genotypes by gene content (the number of alleles that a offered type) is preserved in the ordering of the associated phenotypic values (see <30> because that details). Figure 5 depicts three extreme species of GP maps seen in our simulations. Almost additive GP maps as shown in figure 5A offer VA/VG values very close come one. GP maps with strong magnitude epistasis, but still order-preserving, typically an outcome in intermediary VA/VG values (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 to zero.


Examples of 3 distinct types of genotype-phenotype maps watched in our simulations. Because that each subfigure the phenotypic value is displayed on the y-axis if the x-axises, line colours and also plot columns indicate the genotype in ~ the 3 loci. A. A virtually additive map exemplified through the GP map of the moment to peak concentration of Cdc25 (VA/VG = 0.99) in the cAMP model; B. A totally monotone but non-additive map exemplified through the GP map that the concentration that P2G protein (VA/VG = 0.41) in the glycolysis model; and, C. A strong non-monotonic map is uncovered the time to peak concentration that the pc protein (VA/VG = 0.03) native the circadian model.

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Based on current results from research studies of gene regulation networks <30>, we anticipated that the 3 cGP models with complex regulation involving positive feedback would an outcome in considerably much more non-monotone or order-breaking GP maps 보다 the two pathway models. To check this, we measured the quantity of order-breaking in all simulated GP maps (see Methods) and found a really clear pattern (Figure 6). If the datasets native the glycolysis and also cAMP models included only 1.1% and also 1.3% GP maps with order-breaking for any 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 higher VA/VG worths than non-monotone GP maps because that all five cGP models (Mann-Whitney test; p-values below 1e-10 because that all five models).


The variety of Monte Carlo simulations wherein the GP-map for a offered phenotype is plainly order-breaking (GP maps v N/A>0.05, see Methods) is displayed for the cAMP design (A), the glycolysis design (B), the cabinet cycle design (C), the circadian version (D) and the activity potential version (E). Only phenotypes through at least one Monte carlo simulation causing an order-breaking GP map are shown.

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However, regardless of the reality that the glycolysis design rarely reflects order-breaking even for a single locus, it possesses much lower VA/VG worths than the cAMP model. A plausible explanation is the the steady-state concentration of metabolites have the right to markedly rise for parameter values close to a saddle-node bifurcation point <26>. Simulation outcomes v unstable steady claims were discarded, yet in those instances where among the genotypes (i.e. Parameter sets) come close to the bifurcation suggest without crossing it we obtain genotype-phenotype maps as in number 5B, wherein one genotype (or a little set) provides much greater phenotypic values than the others. Together GP maps, similar to the duplicate factor model in Hill et al. <3>, are recognized to offer low VA/VG ratios despite being monotonic. Similar GP maps offering VA/VG ratios close come zero were also found by Keightley <32> in his study of metabolic models own null alleles at every loci.


Considerations top top the genericity that the results

Our main reason for restricting the sampled genetic variation the parameters to in ~ 30% of the published baseline worths was to stop qualitative (or topological) changes of the dynamics. Such qualitative transforms are regularly biologically realistic descriptions of knockouts or other large genetic changes, for example action potentials of alternative amplitude (alternans) <17>; lose of stable circadian oscillation <23>; and non-viable cell-cycle mutants phenotypes <22>. However, because the heritability and variance component concepts are defined for phenotypes showing consistent rather than discrete variation, we sought to avoid such qualitative transforms here.

We ran simulations with five polymorphic loci for the cAMP (Figure S8A), glycolysis (Figure S8B), cabinet cycle (Figure S9) and activity potential (Figure S10) models (the circadian model defines only three genes explicitly). The result VA/VG values were slightly lower than with 3 loci, but the all at once shape the the distributions and also the clean differences between models did not change. This suggests that ours findings are of basic relevance because that oligogenic traits.

It must be emphasized the the 5 studied cGP models different in number of other elements than those highlighted here, such as the device size (number the state variables) and also the process time scales. These functions could likewise contribute come the it was observed variation in the distributions of VA/VG ratios. However, such structural distinctions are unavoidable when the aim is to to compare experimentally validated models designed come describe specific biological systems. A complementary approach is to examine generic models where mechanism size and equation framework is fixed, if the connectivity matrix can be changed to define a household of solution <35>. This facilitates graph-theoretic comparison of equipment at the expense of some organic realism. Us anticipate the the major conclusions native such studies will be similar to ours, yet it may really well be that other necessary generic insights may likewise come come the fore.

All the models in our study define parts the the moving machinery and the resulting phenotypes are hence cellular quite than organismal. We do not think this is a significant shortcoming in terms of the main conclusions that emerge from our results. However, us anticipate that applications of our method on multiscale models including cellular, tissue and whole-organ phenotypes <36> will carry out a much improved structure for explaining exactly how properties of the GP map vary across and within organic systems in regards to regulatory anatomy and associated hereditary variation <37>, <38>.

As our method can be used in addition to any computational biology model, it has actually the potential to add substantially come a theoretical foundation capable of predicting as soon as we room to mean low or high VA/VG or h2/H2 ratios native the values of regulation biology. Causally cohesive genotype-phenotype modeling thus shows up to qualify as a promising method for complete causal models of organic networks and also physiology through quantitative genes <39>–<44>.


Figure S1.

Flowchart that Monte Carlo simulations and analysis. Flowchart the the Monte Carlo simulations described in the methods section “Monte Carlo simulations” and also subsequent analysis described in the approaches section “Statistical analysis”.

https://doi.org/10.1371/journal.pcbi.1003053.s001

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Figure S2.

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Figure S3.

Graphical depiction of glycolysis model. figure modified native 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 elements where genetic variation was introduced.

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Figure S4.

Graphical depiction of cell cycle model. number modified indigenous the CellML model 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 elements where hereditary variation was introduced.

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Figure S5.

Graphical representation of circadian model. number modified indigenous the CellML design repository (http://models.cellml.org/workspace/leloup_goldbeter_2004). Red numbers, exchange mail to the rows in Table S4, and indicate the model elements where genetic variation to be introduced.

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Figure S6.

Graphical representation of activity potential model. number modified indigenous 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 indicate the model aspects where hereditary variation to be introduced.

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Figure S7.

The empirical cumulative distribution function of VA/VG ratios for phenotypes of the cell cycle model. The empirical cumulative distribution functions (y axis) of VA/VG ratios (x axis) for every phenotypes studied in the cabinet cycle model. The phenotypes are split into 3 groups. Cell occasions refer come the discrete events defined in the model paper and include timing that budding (Bud), time of DNA replication (Rep) and timing the alignment that chromosomes on the metaphase key (Spn). Optimal concentration include the concentration that the phosphorylated anaphase-promoting facility (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 the G1 phase stabilizing protein (Sci1). The moment to optimal phenotypes encompass the time come reach height concentrations the APCP, Cdc6, Clb2, Clb5, Cln2 and also Sci1. Check out Table S8 for further phenotype descriptions and numerical summaries of the distribution of VA/VG ratios.

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Figure S8.

The empirical cumulative distribution role of VA/VG ratios for phenotypes that the cAMP (A) and also the glycolysis (B) models v 5 polymorphic loci. figure 3 shows results indigenous simulations v 3 polymorhpic loci. A. The empirical cumulative circulation functions (y axis) of VA/VG ratios (x axis) for every phenotypes learned in the cAMP model: The initial stable state concentration before adding external glucose that 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 adding glucose of these proteins (cAMPv, Ras2av, Cdc25v and PKAiv), the Kelch repeat homologue protein (Krhv), the G-protein Gpa2a (Gpa2av), and also the phosphodiesterase (Pde1v). The time required to reach the peak values (cAMPt, Ras2at, Cdc25t, PKAit, Krht, Gpa2at, Ped1t). B. The empirical cumulative distribution role (y axis) that VA/VG ratios (x axis) because that the secure state concentrations 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 adenin nucleotide (P), 2-phosphoglyerate (P2G), 3-phosphoglycerate (P3G), phosphoenolpyruvate (PEP), pyruvate (PYP), and trio-phosphate (TRIO).

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Figure S9.

The empirical accumulation distribution duty of VA/VG ratios because that phenotypes of the cell cycle model with 5 polymorphic loci. figure S7 mirrors results from simulations v 3 polymorhpic loci. The empirical cumulative circulation functions (y axis) that VA/VG ratios (x axis) for every phenotypes studied in the cell cycle model. The phenotypes are divided into 3 groups. Cell events refer come the discrete events identified in the model paper and encompass timing the budding (Bud), time of DNA replication (Rep) and timing the alignment the chromosomes on the metaphase bowl (Spn). Top concentration incorporate the concentration that 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 the G1 step stabilizing protein (Sci1). The time to top phenotypes incorporate the time to reach optimal concentrations of APCP, Cdc6, Clb2, Clb5, Cln2 and also Sci1.

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Figure S10.

The empirical cumulative distribution role of VA/VG ratios for phenotypes the the action potential model with 5 polymorphic loci. figure 4B shows results native simulations through 3 polymorhpic loci. The empirical cumulative circulation functions (y axis) the VA/VG ratios (x axis) for phenotypes studied in the action potential model: time come 25%, 50%, 75% and 90% of early stage values, the amplitude, initial values (Base), top values, time to reach optimal values of action potential (left panel) and calcium transient (right panel) are shown.

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Table S1.

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Table S2.

Polymorphic model facets of the glycolysis model. A list of glycolysis version elements and parameters provided to manifest genetic variation. Parameter names native the initial publication <20>, names offered in the SBML document retrieved from http://www.ebi.ac.uk/biomodels-main/BIOMD0000000064 and also baseline values through units.

https://doi.org/10.1371/journal.pcbi.1003053.s012

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Table S3.

Polymorphic model aspects of the cell cycle model. A perform of cell cycle version elements and also parameters used to manifest hereditary variation. Parameter names native Table 1 and Table 2 in the original publication <22>, names used in the CellML document retrieved native http://models.cellml.org/workspace/chen_calzone_csikasznagy_cross_novak_tyson_2004 and also baseline values through units.

https://doi.org/10.1371/journal.pcbi.1003053.s013

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Table S4.

Polymorphic model aspects of the circadian model. A list of circadian design elements and also parameters offered to manifest genetic variation. Parameter names indigenous Table 1 (parameter collection 4) in the initial publication <23>, names offered in the CellML document “leloup_goldbeter_2004.cellml” retrieved indigenous http://models.cellml.org/workspace/leloup_goldbeter_2004/ and baseline values v units.

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Table S5.

Polymorphic model elements of the action potential model. A perform of activity potential design elements and parameters provided to manifest genetic variation. Parameter names native Table B1 in the original publication <24>, names provided in the CellML file which is available as supplementary product (filename “LNCS model.zip”) in ~ doi:10.3389/fphys.2011.00106 and baseline values with units.

https://doi.org/10.1371/journal.pcbi.1003053.s015

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Table S6.

Summary the phenotype descriptions, variability thresholds and also distribution of VA/VG ratios because that the cAMP model. The an initial three columns list the phenotype abbreviations offered in this study, a text description of the phenotypes and also their units. The thresholds provided to filter the end dataset with really low relative and/or pure variability are detailed in the next two columns, followed by the variety of Monte Carlo simulations (out that 1000) passing the threshold. The last 7 columns contain quantiles and method of the VA/VG worths for the datasets pass the variability threshold.

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Table S7.

Summary of phenotype descriptions, variability thresholds and distribution of VA/VG ratios for the glycolysis model. The first three columns perform the phenotype abbreviations offered in this study, a text summary of the phenotypes and also their units. The thresholds provided to filter out dataset with an extremely low family member and/or pure variability are noted in the following two columns, adhered to by the variety of Monte Carlo simulations (out the 1000) pass the threshold. The last 7 columns contain quantiles and method of the VA/VG worths for the datasets pass the variability threshold.

https://doi.org/10.1371/journal.pcbi.1003053.s017

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Table S8.

Summary the phenotype descriptions, variability thresholds and also distribution the VA/VG ratios for the cabinet cycle model. The an initial three columns list the phenotype abbreviations supplied in this study, a text description of the phenotypes and their units. The thresholds offered to filter the end dataset with very low loved one and/or pure variability are provided in the following two columns, adhered to by the variety of Monte Carlo simulations (out the 1000) passing the threshold. The critical 7 columns save quantiles and means of the VA/VG worths for the datasets passing the variability threshold.

https://doi.org/10.1371/journal.pcbi.1003053.s018

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Table S9.

Summary that phenotype descriptions, variability thresholds and distribution that VA/VG ratios because that 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 provided to filter out dataset with an extremely low loved one and/or absolute variability are provided in the next two columns, adhered to by the variety of Monte Carlo simulations (out of 1000) pass the threshold. The critical 7 columns save quantiles and method of the VA/VG values for the datasets pass the variability threshold. Abbreviations: phosphorylated – phos., cytosolic – cyt., atom – nuc., bottom concentration – b.c., optimal concentration – p.c.

https://doi.org/10.1371/journal.pcbi.1003053.s019

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Table S10.

Summary the phenotype descriptions, variability thresholds and also distribution the VA/VG ratios for the action potential model. The first three columns perform the phenotype abbreviations used in this study, a text description of the phenotypes and also their units. The thresholds provided to filter the end dataset with really low loved one and/or pure variability are detailed in the following two columns, adhered to by the number of Monte Carlo simulations (out that 1000) passing the threshold. The critical 7 columns save quantiles and method of the VA/VG worths for the datasets passing the variability threshold.

https://doi.org/10.1371/journal.pcbi.1003053.s020

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Text S1.

More in-depth descriptions that the five cGP models.

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Acknowledgments

We room thankful come Katherine C. Chen for assist on implementing and solving the cell cycle model.

See more: What Does Hi De Hi De Hi De Ho, Cab Calloway And His Orchestra


Author Contributions

Conceived the study: ABG SWO. Performed simulations and also analysis: YW ABG JOV. Created the paper: YW JOV SWO ABG.


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