BACKGROUND
technological developments continue to dramatically expand the enterprise of dna sequencing.
in particular, the emergence of so-called "next-generation" instruments  is opening a new chapter of genomic research  <cit> .
if we characterize sequencing economy by the ratio of project speed to total project cost, ngis are orders of magnitude superior to their traditional sanger-based predecessors.
indeed, they are the first systems to demonstrate the economic feasibility of sequencing individual genomes on a large scale  <cit> .
future efforts will undoubtedly use ngis to address issues in medical sequencing and personal genomics  <cit> , but these instruments are also poised for major contributions at the population level  <cit> .
for example, the thousand genomes project  is focusing on comprehensive identification of variants in the human population through cohort-level whole-genome sequencing using ngis  <cit> .
one of its main goals is to discover and characterize rare single nucleotide alleles, basically those present at minor allele frequencies around 1% or less.
this region was not accessible to the earlier hapmap project  <cit> .
rarer instances are obviously much more difficult to find and necessitate gathering enormously larger amounts of data.
such demands will obviously extend to any future such projects one might envision, including those for model organisms, agriculturally important species, cancer genomes, infectious agents, etc.
the success of such variation projects depends upon adequately understanding the relevant process engineering issues and subsequently crafting a suitable project design.
one concern in traditional single-genome sequencing is the so-called "stopping problem"  <cit> ], which is the proposition of estimating what redundancy will suffice for a desired level of genomic coverage.
variation projects similarly require specification of a total, project-wide redundancy, r. yet, because they necessarily involve multiple genomes, an essentially new design question also emerges.
that is, how does one optimize the number of samples, σ, versus the redundancy allotted per sample, ρ, such that the probability of finding a rare variant, pv, is maximized?
the existence of such optima is intuitively clear.
heavily sequencing only a few samples will tend to miss a variant because it is unlikely to be present in the original sample set.
conversely, light sequencing of too many samples may overlook the variant by virtue of insufficient coverage for any samples actually harboring it.
somewhere between these extremes lie optimum combinations of parameters.
at present, this issue can only be addressed in ad hoc, fairly inefficient ways.
for example, the tgp conducted both painstaking computer simulations and pilot sequencing phases involving hundreds of genomes to aid in designing the full-scale project  <cit> .
while certainly informative, even such seemingly extensive data may not, by themselves, give a complete picture of optimization because combinations of the many underlying variables  lead to an enormous solution space.
we comment further on this aspect below.
existing theory is also ineffective because sequence coverage has not yet been considered  <cit> .
†some variables are modified with a "star" superscript to denote optima, for example σ* is the optimum sample size for a project, ρ* the optimum per-sample redundancy, and  the discovery probability under optimal conditions.
here, we examine optimization from a more focused mathematical perspective.
our treatment accounts for sequence errors via the proxy of a variable read covering count  <cit> , but it omits secondary, project-specific details like software idiosyncrasies  <cit> , instrument-specific biases  <cit> , and alignment issues  <cit> .
the solution leads to a set of general, though unexpectedly simple optimization principles, which correct some earlier speculation  <cit>  and are useful as first approximations for actual projects.
because these rules appreciably narrow the solution space, they also offer good starting points for even more targeted numerical and empirical searches that might account for secondary effects, if such are deemed necessary.
RESULTS
the term "rare variant" is routinely taken to mean a rare allele, although it can also mean a rare snp genotype.
take ϕ to be the variant frequency, i.e.
the minor allele frequency or the rare homozygous genotype frequency, as appropriate.
we assume the tgp convention whereby samples are sequenced separately to uniform depths  <cit> , instead of being pooled first.
the general theory then encompasses the multiple-genome population sequencing problem and its subsequent design optimization.
analytical characterization of discovery in multiple genomes
theorem  <dig> .
let da be the event that a rare allele is detected, i.e.
found by the investigator in a sequenced diploid genome sample.
its probability is
where
is the coverage probability of spanning the allele's genomic position on a chromosome with at least τ sequence reads.
let σ independent, randomly-selected samples each be sequenced uniformly to haploid depth ρ. then, if k is the random variable representing the number of samples the variant is found in and if n is the minimum number of observations necessary to declare the variant as being "discovered", the discovery event is defined as k ≥ n and its probability is
theorem  <dig> .
the probability of dg, the event that a rare genotype is detected in a sample, is
and its discovery probability is again given by eq.
<dig>  except where dg, replaces da.
statement of the optimization problem
variant discovery is a constrained optimization problem  <cit> , which can be stated as follows.
given the biological parameter ϕ and project-specific design parameters r, σ, τ, pv, min, and n, maximize the objective function pv, subject to both the equality constraint
and to the auxiliary constraint
in practical terms, we want to most efficiently discover a variant at the lowest possible cost, as represented by r.
although the problem is framed in terms of finding a single variant, actual projects are apt to be specified according to discovering a certain average number of rare variants.
these scenarios are equivalent, as eq.
<dig> also quantifies the expected fraction of variants that will be found in the project.
for example, pv, min =  <dig>  indicates finding 95%, on average, of the variants occurring at some value of ϕ.
optimizing for single and double variant observations
leaving aside the optimization of ρ versus σ for a moment, the least obvious of the project-specific parameters to specify is arguably n. higher values may exceed the actual number of instances in the sample set, resulting in a priori failure of the project.
we will therefore concentrate on the experimentally relevant special cases n =  <dig> and n =  <dig>  the former is clearly a minimum requirement, while the latter serves to better discern between a rare population variant and a snp that is unique to an individual sample .
because we have an explicitly-defined equality constraint in the form of eq.
<dig>  the number of design variables can be reduced by one  <cit> .
specifically, substituting ρ = r/σ into eq.
<dig> allows us to write a constrained form of the coverage probability, which in turn furnishes constrained expressions for the probabilities of events da and dg.
it is expedient at this point to switch from the event-based notation of probability used up until now to the eulerian  convention for the calculus-based aspect of optimization.
specifically, let fτ, iwith i ∈ {a, g} represent the now-constrained probabilities of da and dg.
we now state the following important optimization conditions.
theorem  <dig> .
the optimum number of samples in a multiple-genome variation project for n =  <dig> is governed by the differential equation
and for n =  <dig> by the differential equation
in particular, the roots of these equations in σ indicate maxima in pv for rare alleles and genotypes.
each setting of the independent variables has one such optimum, σ*, which is necessarily a global optimum.
discussion
finding rare variants is clearly an important aspect of both population and medical genetics  <cit> .
the discovery process was not feasible before the advent of ngis, but is now being actively prototyped through efforts like the tgp  <cit>  and will likely become more routine in the future.
this eventuality motivates examination of the problem from a general perspective, similar in spirit to theoretical treatments of single genomes  <cit> .
the following sections report on both some of the broad trends across the design variable spectrum, as well as optimal conditions for the important special cases of n =  <dig> and n =  <dig>
general trends
fig.
<dig> shows pv versus σ for variants appearing at 1% frequency for thresholds of n =  <dig> and τ =  <dig>  the latter appears to have emerged as the de facto choice to better control for sequencing errors  <cit> .
aside from the expected trend that performance improves as more data are gathered, the curves show two notable properties.
first, σ*, the sample size at which the maximum pv occurs, increases with the project redundancy.
this dependence means that a project cannot generally be optimized by selecting σ in advance of other factors.
put another way, outright specification of σ almost certainly assures that the discovery process will not be optimal.
we expand further upon this point below.
fig.
<dig> also shows that curves are not symmetric with respect to σ*.
the rate of drop-off of pv for a given project-wide redundancy is much more severe for σ <σ*, implying that it is better to err in sequencing too many samples rather than too few.
it is interesting to examine one of the tgp sequencing pilot phases in this context, which specifies 2× data collection for each of σ =  <dig> samples  <cit> .
here, r = 2· <dig> =  <dig>  which is one of the curves plotted in fig.
<dig>  using the above thresholds, this design yields pv ≈ 61%, whereas the optimal configuration returns pv ≈ 66% for only about  <dig> samples.
despite using almost twice as many samples as is optimal, this design remains relatively good, precisely because of the non-symmetric behavior.
constant sample-size designs and the stalling effect
the above discussion suggests that investigators should consider abandoning the idea of choosing σ outright.
an earlier projection offers an interesting case study to further illustrate this point.
gibbs  <cit>  postulated that σ =  <dig>  samples would be a good way of discovering extremely rare variants occurring at  <dig> %.
fig.
<dig> shows the implications of such a σ-based design.
as r increases, σ* marches to the right on the abscissa, eventually passing through the pre-selected σ =  <dig>  at around r =  <dig> .
it continues rightward, leaving our fixed sample datum in the left-side wake of the optimum , where the associated probability is now heavily penalized.
in fact, the probability stalls at a value of roughly pv ≈  <dig> , regardless of the amount of additional data poured into the project.
although this stalling effect may initially seem counter-intuitive, its explanation is quite straightforward.
if we hold σ fixed while letting r increase without bounds, then ρ also grows without bounds .
in the limit, each sample will be perfectly sequenced, i.e.
p →  <dig> in eq.
<dig>  discovery is then simply a function of whether or not the variant is present in the original sample set.
if so, it is absolutely certain to be discovered.
the corresponding probabilities are then simple special cases of the model in thms.
<dig> and  <dig>  for example, for n =  <dig> observation of a rare allele we find
which is asymptotically identical to what is obtained if coverage is not considered at all  <cit> .
the basic problem associated with constant sample-size designs is immediately apparent in this equation.
given small ϕ, the exponential term decays very slowly and can only be compensated for by increasing σ. the challenge, of course, is to do this such that pv attains a maximum.
remarks on optimization methods
we commented above that empirical prototyping and numerical simulation are unlikely to give complete insights to the general optimization problem because of the size of the solution space.
consider that the relationship between two parameters requires only a single curve on an x-y plot, three parameters require a family of curves on one plot, four a textbook of family-type plots, and so forth.
richard bellman, who developed the optimization technique of dynamic programming, called this phenomenon the "curse of dimensionality".
table  <dig> shows that we have  <dig> variables in our particular problem, however, even this is somewhat misleading because it does not consider the probabilistic nature of the problem.
that is, pv can only be established as an expected value through a sufficient number of repeated trials for each particular combination of the independent variables.
this is the basic tactic used in simulation.
the population model in thms.
<dig> and  <dig> improves matters considerably, furnishing pv explicitly in terms of .
one could march through every combination of these variables, evaluating pv for each, and log maxima that attain given levels of pv, min.
though this approach would be enormously more efficient than naïve brute-force simulation, the calculations needed to adequately survey the floating-point "continuum" of the real-valued variables remain basically infeasible.
consequently, we still might not expect to discern any latent general laws.
the weak optimization problem
we resort instead to thm.
<dig>  whose roots for n =  <dig> and n =  <dig> represent optimal sample sizes, σ*.
let us first describe some unexpected properties found among the independent variables.
these are important in that they furnish a direct solution to what might be called the weak optimization problem.
this is the proposition that relaxes the condition defined by pv, min.
in effect, weak optimization provides the optimal probability, , subject to a pre-determined r rather than a given pv, min >  <dig>
fig.
<dig> shows σ* versus r for representative parameter settings.
data collapse onto curves according to variant type.
in each case, σ* = σ* and σ* ∝ r. these observations, coupled with σ* = r/ρ* from eq.
<dig> then imply σ* = r/ρ*.
in other words, ρ* is only a function of τ .
this is quite a significant finding because it immediately establishes the best sample redundancy for a project.
in essence, this observation indicates that optimizable designs for rare variants are based on constant values of ρ rather than constant values of σ  <cit> .
†see eqs.
<dig>   <dig>  16
we emphasize that the numbers in table  <dig> are not based on first-principles and are not strictly encoded in the governing equations.
rather, they are fortuitous empiricisms, restricted to the parameter values that characterize rare-variant projects.
fig.
<dig> demonstrates that, while ρ* does indeed only depend upon τ up to allele frequencies of about 1%, it becomes a more complicated function of additional variables for higher frequencies.
remarks on the special case of τ =  <dig> for rare alleles
the case of τ =  <dig> is conspicuously absent for rare alleles in figs.
<dig> and  <dig> because its optimum sample size is not finite.
unlike the other cases, pv approaches its maximum as σ → ∞, for example
here, we have the seemingly contradictory implication that we should spread a finite amount of sequence resources over the largest number of samples, each of which will then have a vanishingly small ρ. mathematically speaking, the rate by which the per-sample f <dig>  a decreases precisely offsets the favorable rate of increasing sample size, whereby pv does not asymptotically vanish.
however, there will usually be good secondary reasons for limiting σ in practice, e.g.
cost of sample procurement.
moreover, conditions approach the limiting value rather quickly, for example setting ρ = r/σ ≤  <dig>  brings pv very close to the expression in eq.
<dig>  r is the main factor governing discovery under these conditions and its value can be calculated for a desired pv by simply inverting eq.
<dig>
optimal designs for single and double variant observations
the weak solution specifies constants of ρ* , which simultaneously imply σ* for any choice of r. these properties subsequently fix certain relationships within the general problem, so that optimization for a desired pv, min in eq.
<dig> reduces to the task of solving directly for ϕ .
fig.
<dig> shows the resulting optimal designs for τ =  <dig>  a setting characteristic of recent projects  <cit> .
results are plotted for pv, min = 95%, the same threshold set by the tgp.
all curves show a special kind of log-log relationship between ϕ and r* in which the slope is - <dig>  in other words, optimal designs can be expressed as a family of log-log curves having the form ϕ r* = c, where c is a so-called optimization coefficient for each combination of the variables.
of course, knowing c immediately enables one to compute r* and subsequently σ* = r*/ρ* for a desired ϕ, which is of enormous practical value for project design.
table  <dig> shows c for the configurations having well-defined optimum redundancies, although we note that eq.
<dig> also follows this form, having c =  <dig> .
r* is indicative of the total resources a project requires, so c is also useful in comparing relative costs.
for example, requiring two observations of a rare allele instead of just one would only be, somewhat counter-intuitively, about 60% more expensive if both schemes were to be done optimally.
consider the example of the tgp, whose sizable ad hoc design effort was already mentioned above.
for n = τ =  <dig> at the 95% level, table  <dig> indicates c =  <dig> .
assuming 1% rare allele discovery, optimal processing calls for roughly  <dig> samples sequenced to  <dig> × each, for a project total of r = 1580×.
given the long-standing convention of specifying ρ in whole units, these results largely confirm the tgp design, though in a more precise fashion.
that is, tgp has only winnowed the sample size to 400- <dig> per population cluster, with each sample sequenced to 4×  <cit> .
the associated pv curve is relatively flat in  <dig> ≤ σ ≤  <dig>  but this imprecision, coupled with a round value of ρ, still imposes a degree of cost liability.
for instance, on the outer end, the project would expend 4· <dig> = 2000× in data, roughly 25% more than that required for 95% confidence.
project modifications are readily analyzed, for example, reaching alleles down to ϕ =  <dig> % would simply require doubling the project: about  <dig> samples with r = 3160×.
analysis of genotypes is now similarly trivial.
CONCLUSIONS
sequence variation is often called the "currency" of genetics  <cit>  and whole-genome sequence variation projects, enabled by continuing advances in technology, will likely become both increasingly important and routine in biomedical research.
although finding common occurrences is no longer considered to be very difficult, rare ones remain challenging because of the significantly larger amounts of data that must be gathered.
process optimization has to be considered much more carefully here.
we have reported a general, though remarkably simple set of optimization principles based on analyzing the population sequencing problem.
results largely confirm the design of a special case, that of the tgp, but also permit immediate analysis of a much broader set of possible project requirements.
derivation of optimal conditions for even higher n and/or τ would be a mechanical, albeit not entirely trivial extension of the mathematics shown here, but the experimental feasibility of such designs for future projects remains unclear.
population structure is another consideration, as rare variants are likely to be associated with particular geographic regions and their sub-populations  <cit> .
a few issues are relevant here.
first, some studies target the variation underlying specific phenotypes  <cit> , but variant discovery projects do not place strong preference on the kinds of variation that are sought.
second, ρ* is not a function of rareness , meaning that latent population-related differences in frequency will not ruin optimality.
one should simply treat each desired sub-population separately, making no differential adjustments to per-sample redundancies.
this strategy assures discovery of population-specific variants and, incidentally, is precisely what the tgp is following.

 technological developments continue to dramatically expand the enterprise of dna sequencing. in particular, the emergence of so-called "next-generation" instruments is opening a new chapter of genomic research. if we characterize sequencing economy by the ratio of project speed to total project cost, ngis are orders of magnitude superior to their traditional sanger-based predecessors. indeed, they are the first systems to demonstrate the economic feasibility of sequencing individual genomes on a large scale. future efforts will undoubtedly use ngis to address issues in medical sequencing and personal genomics, but these instruments are also poised for major contributions at the population level. for example, the thousand genomes project is focusing on comprehensive identification of variants in the human population through cohort-level whole-genome sequencing using ngis. one of its main goals is to discover and characterize rare single nucleotide alleles, basically those present at minor allele frequencies around 1% or less. this region was not accessible to the earlier hapmap project. rarer instances are obviously much more difficult to find and necessitate gathering enormously larger amounts of data. such demands will obviously extend to any future such projects one might envision, including those for model organisms, agriculturally important species, cancer genomes, infectious agents, etc. the success of such variation projects depends upon adequately understanding the relevant process engineering issues and subsequently crafting a suitable project design. one concern in traditional single-genome sequencing is the so-called "stopping problem" ], which is the proposition of estimating what redundancy will suffice for a desired level of genomic coverage. variation projects similarly require specification of a total, project-wide redundancy, r. yet, because they necessarily involve multiple genomes, an essentially new design question also emerges. that is, how does one optimize the number of samples, σ, versus the redundancy allotted per sample, ρ, such that the probability of finding a rare variant, pv, is maximized? the existence of such optima is intuitively clear. heavily sequencing only a few samples will tend to miss a variant because it is unlikely to be present in the original sample set. conversely, light sequencing of too many samples may overlook the variant by virtue of insufficient coverage for any samples actually harboring it. somewhere between these extremes lie optimum combinations of parameters. at present, this issue can only be addressed in ad hoc, fairly inefficient ways. for example, the tgp conducted both painstaking computer simulations and pilot sequencing phases involving hundreds of genomes to aid in designing the full-scale project. while certainly informative, even such seemingly extensive data may not, by themselves, give a complete picture of optimization because combinations of the many underlying variables lead to an enormous solution space. we comment further on this aspect below. existing theory is also ineffective because sequence coverage has not yet been considered. †some variables are modified with a "star" superscript to denote optima, for example σ* is the optimum sample size for a project, ρ* the optimum per-sample redundancy, and the discovery probability under optimal conditions. here, we examine optimization from a more focused mathematical perspective. our treatment accounts for sequence errors via the proxy of a variable read covering count, but it omits secondary, project-specific details like software idiosyncrasies, instrument-specific biases, and alignment issues. the solution leads to a set of general, though unexpectedly simple optimization principles, which correct some earlier speculation and are useful as first approximations for actual projects. because these rules appreciably narrow the solution space, they also offer good starting points for even more targeted numerical and empirical searches that might account for secondary effects, if such are deemed necessary.  the term "rare variant" is routinely taken to mean a rare allele, although it can also mean a rare snp genotype. take ϕ to be the variant frequency, i.e. the minor allele frequency or the rare homozygous genotype frequency, as appropriate. we assume the tgp convention whereby samples are sequenced separately to uniform depths, instead of being pooled first. the general theory then encompasses the multiple-genome population sequencing problem and its subsequent design optimization. analytical characterization of discovery in multiple genomes theorem. let da be the event that a rare allele is detected, i.e. found by the investigator in a sequenced diploid genome sample. its probability is where is the coverage probability of spanning the allele's genomic position on a chromosome with at least τ sequence reads. let σ independent, randomly-selected samples each be sequenced uniformly to haploid depth ρ. then, if k is the random variable representing the number of samples the variant is found in and if n is the minimum number of observations necessary to declare the variant as being "discovered", the discovery event is defined as k ≥ n and its probability is theorem. the probability of dg, the event that a rare genotype is detected in a sample, is and its discovery probability is again given by eq. except where dg, replaces da. statement of the optimization problem variant discovery is a constrained optimization problem, which can be stated as follows. given the biological parameter ϕ and project-specific design parameters r, σ, τ, pv, min, and n, maximize the objective function pv, subject to both the equality constraint and to the auxiliary constraint in practical terms, we want to most efficiently discover a variant at the lowest possible cost, as represented by r. although the problem is framed in terms of finding a single variant, actual projects are apt to be specified according to discovering a certain average number of rare variants. these scenarios are equivalent, as eq. also quantifies the expected fraction of variants that will be found in the project. for example, pv, min = indicates finding 95%, on average, of the variants occurring at some value of ϕ. optimizing for single and double variant observations leaving aside the optimization of ρ versus σ for a moment, the least obvious of the project-specific parameters to specify is arguably n. higher values may exceed the actual number of instances in the sample set, resulting in a priori failure of the project. we will therefore concentrate on the experimentally relevant special cases n = and n = the former is clearly a minimum requirement, while the latter serves to better discern between a rare population variant and a snp that is unique to an individual sample. because we have an explicitly-defined equality constraint in the form of eq. the number of design variables can be reduced by one. specifically, substituting ρ = r/σ into eq. allows us to write a constrained form of the coverage probability, which in turn furnishes constrained expressions for the probabilities of events da and dg. it is expedient at this point to switch from the event-based notation of probability used up until now to the eulerian convention for the calculus-based aspect of optimization. specifically, let fτ, iwith i ∈ {a, g} represent the now-constrained probabilities of da and dg. we now state the following important optimization conditions. theorem. the optimum number of samples in a multiple-genome variation project for n = is governed by the differential equation and for n = by the differential equation in particular, the roots of these equations in σ indicate maxima in pv for rare alleles and genotypes. each setting of the independent variables has one such optimum, σ*, which is necessarily a global optimum. discussion finding rare variants is clearly an important aspect of both population and medical genetics. the discovery process was not feasible before the advent of ngis, but is now being actively prototyped through efforts like the tgp and will likely become more routine in the future. this eventuality motivates examination of the problem from a general perspective, similar in spirit to theoretical treatments of single genomes. the following sections report on both some of the broad trends across the design variable spectrum, as well as optimal conditions for the important special cases of n = and n = general trends fig. shows pv versus σ for variants appearing at 1% frequency for thresholds of n = and τ = the latter appears to have emerged as the de facto choice to better control for sequencing errors. aside from the expected trend that performance improves as more data are gathered, the curves show two notable properties. first, σ*, the sample size at which the maximum pv occurs, increases with the project redundancy. this dependence means that a project cannot generally be optimized by selecting σ in advance of other factors. put another way, outright specification of σ almost certainly assures that the discovery process will not be optimal. we expand further upon this point below. fig. also shows that curves are not symmetric with respect to σ*. the rate of drop-off of pv for a given project-wide redundancy is much more severe for σ <σ*, implying that it is better to err in sequencing too many samples rather than too few. it is interesting to examine one of the tgp sequencing pilot phases in this context, which specifies 2× data collection for each of σ = samples. here, r = 2· = which is one of the curves plotted in fig. using the above thresholds, this design yields pv ≈ 61%, whereas the optimal configuration returns pv ≈ 66% for only about samples. despite using almost twice as many samples as is optimal, this design remains relatively good, precisely because of the non-symmetric behavior. constant sample-size designs and the stalling effect the above discussion suggests that investigators should consider abandoning the idea of choosing σ outright. an earlier projection offers an interesting case study to further illustrate this point. gibbs postulated that σ = samples would be a good way of discovering extremely rare variants occurring at %. fig. shows the implications of such a σ-based design. as r increases, σ* marches to the right on the abscissa, eventually passing through the pre-selected σ = at around r =. it continues rightward, leaving our fixed sample datum in the left-side wake of the optimum, where the associated probability is now heavily penalized. in fact, the probability stalls at a value of roughly pv ≈, regardless of the amount of additional data poured into the project. although this stalling effect may initially seem counter-intuitive, its explanation is quite straightforward. if we hold σ fixed while letting r increase without bounds, then ρ also grows without bounds. in the limit, each sample will be perfectly sequenced, i.e. p → in eq. discovery is then simply a function of whether or not the variant is present in the original sample set. if so, it is absolutely certain to be discovered. the corresponding probabilities are then simple special cases of the model in thms. and for example, for n = observation of a rare allele we find which is asymptotically identical to what is obtained if coverage is not considered at all. the basic problem associated with constant sample-size designs is immediately apparent in this equation. given small ϕ, the exponential term decays very slowly and can only be compensated for by increasing σ. the challenge, of course, is to do this such that pv attains a maximum. remarks on optimization methods we commented above that empirical prototyping and numerical simulation are unlikely to give complete insights to the general optimization problem because of the size of the solution space. consider that the relationship between two parameters requires only a single curve on an x-y plot, three parameters require a family of curves on one plot, four a textbook of family-type plots, and so forth. richard bellman, who developed the optimization technique of dynamic programming, called this phenomenon the "curse of dimensionality". table shows that we have variables in our particular problem, however, even this is somewhat misleading because it does not consider the probabilistic nature of the problem. that is, pv can only be established as an expected value through a sufficient number of repeated trials for each particular combination of the independent variables. this is the basic tactic used in simulation. the population model in thms. and improves matters considerably, furnishing pv explicitly in terms of. one could march through every combination of these variables, evaluating pv for each, and log maxima that attain given levels of pv, min. though this approach would be enormously more efficient than naïve brute-force simulation, the calculations needed to adequately survey the floating-point "continuum" of the real-valued variables remain basically infeasible. consequently, we still might not expect to discern any latent general laws. the weak optimization problem we resort instead to thm. whose roots for n = and n = represent optimal sample sizes, σ*. let us first describe some unexpected properties found among the independent variables. these are important in that they furnish a direct solution to what might be called the weak optimization problem. this is the proposition that relaxes the condition defined by pv, min. in effect, weak optimization provides the optimal probability,, subject to a pre-determined r rather than a given pv, min > fig. shows σ* versus r for representative parameter settings. data collapse onto curves according to variant type. in each case, σ* = σ* and σ* ∝ r. these observations, coupled with σ* = r/ρ* from eq. then imply σ* = r/ρ*. in other words, ρ* is only a function of τ. this is quite a significant finding because it immediately establishes the best sample redundancy for a project. in essence, this observation indicates that optimizable designs for rare variants are based on constant values of ρ rather than constant values of σ. †see eqs. 16 we emphasize that the numbers in table are not based on first-principles and are not strictly encoded in the governing equations. rather, they are fortuitous empiricisms, restricted to the parameter values that characterize rare-variant projects. fig. demonstrates that, while ρ* does indeed only depend upon τ up to allele frequencies of about 1%, it becomes a more complicated function of additional variables for higher frequencies. remarks on the special case of τ = for rare alleles the case of τ = is conspicuously absent for rare alleles in figs. and because its optimum sample size is not finite. unlike the other cases, pv approaches its maximum as σ → ∞, for example here, we have the seemingly contradictory implication that we should spread a finite amount of sequence resources over the largest number of samples, each of which will then have a vanishingly small ρ. mathematically speaking, the rate by which the per-sample f a decreases precisely offsets the favorable rate of increasing sample size, whereby pv does not asymptotically vanish. however, there will usually be good secondary reasons for limiting σ in practice, e.g. cost of sample procurement. moreover, conditions approach the limiting value rather quickly, for example setting ρ = r/σ ≤ brings pv very close to the expression in eq. r is the main factor governing discovery under these conditions and its value can be calculated for a desired pv by simply inverting eq. optimal designs for single and double variant observations the weak solution specifies constants of ρ*, which simultaneously imply σ* for any choice of r. these properties subsequently fix certain relationships within the general problem, so that optimization for a desired pv, min in eq. reduces to the task of solving directly for ϕ. fig. shows the resulting optimal designs for τ = a setting characteristic of recent projects. results are plotted for pv, min = 95%, the same threshold set by the tgp. all curves show a special kind of log-log relationship between ϕ and r* in which the slope is - in other words, optimal designs can be expressed as a family of log-log curves having the form ϕ r* = c, where c is a so-called optimization coefficient for each combination of the variables. of course, knowing c immediately enables one to compute r* and subsequently σ* = r*/ρ* for a desired ϕ, which is of enormous practical value for project design. table shows c for the configurations having well-defined optimum redundancies, although we note that eq. also follows this form, having c =. r* is indicative of the total resources a project requires, so c is also useful in comparing relative costs. for example, requiring two observations of a rare allele instead of just one would only be, somewhat counter-intuitively, about 60% more expensive if both schemes were to be done optimally. consider the example of the tgp, whose sizable ad hoc design effort was already mentioned above. for n = τ = at the 95% level, table indicates c =. assuming 1% rare allele discovery, optimal processing calls for roughly samples sequenced to × each, for a project total of r = 1580×. given the long-standing convention of specifying ρ in whole units, these results largely confirm the tgp design, though in a more precise fashion. that is, tgp has only winnowed the sample size to 400- per population cluster, with each sample sequenced to 4×. the associated pv curve is relatively flat in ≤ σ ≤ but this imprecision, coupled with a round value of ρ, still imposes a degree of cost liability. for instance, on the outer end, the project would expend 4· = 2000× in data, roughly 25% more than that required for 95% confidence. project modifications are readily analyzed, for example, reaching alleles down to ϕ = % would simply require doubling the project: about samples with r = 3160×. analysis of genotypes is now similarly trivial.  sequence variation is often called the "currency" of genetics and whole-genome sequence variation projects, enabled by continuing advances in technology, will likely become both increasingly important and routine in biomedical research. although finding common occurrences is no longer considered to be very difficult, rare ones remain challenging because of the significantly larger amounts of data that must be gathered. process optimization has to be considered much more carefully here. we have reported a general, though remarkably simple set of optimization principles based on analyzing the population sequencing problem. results largely confirm the design of a special case, that of the tgp, but also permit immediate analysis of a much broader set of possible project requirements. derivation of optimal conditions for even higher n and/or τ would be a mechanical, albeit not entirely trivial extension of the mathematics shown here, but the experimental feasibility of such designs for future projects remains unclear. population structure is another consideration, as rare variants are likely to be associated with particular geographic regions and their sub-populations. a few issues are relevant here. first, some studies target the variation underlying specific phenotypes, but variant discovery projects do not place strong preference on the kinds of variation that are sought. second, ρ* is not a function of rareness, meaning that latent population-related differences in frequency will not ruin optimality. one should simply treat each desired sub-population separately, making no differential adjustments to per-sample redundancies. this strategy assures discovery of population-specific variants and, incidentally, is precisely what the tgp is following. 
we pose and solve the population sequencing design problem and subsequently apply standard optimization techniques that maximize the discovery probability. specifically, optimal processing occurs at constant values of the per-sample redundancy, refuting current notions that sample size should be selected outright. optimal project-wide redundancy and sample size are then shown to be inversely proportional to the desired variant frequency. our results largely concur with the empirical design of the thousand genomes project, though they furnish some additional refinement. the optimization principles reported here dramatically simplify the design process and should be broadly useful as rare-variant projects become both more important and routine in the future. we find that parameter values characteristic of rare-variant projects lead to a general, yet remarkably simple set of optimization rules.
 rare population variants are known to have important biomedical implications, but their systematic discovery has only recently been enabled by advances in dna sequencing. the design process of a discovery project remains formidable, being limited to ad hoc mixtures of extensive computer simulation and pilot sequencing. here, the task is examined from a general mathematical perspective.  we pose and solve the population sequencing design problem and subsequently apply standard optimization techniques that maximize the discovery probability. emphasis is placed on cases whose discovery thresholds place them within reach of current technologies. we find that parameter values characteristic of rare-variant projects lead to a general, yet remarkably simple set of optimization rules. specifically, optimal processing occurs at constant values of the per-sample redundancy, refuting current notions that sample size should be selected outright. optimal project-wide redundancy and sample size are then shown to be inversely proportional to the desired variant frequency. a second family of constants governs these relationships, permitting one to immediately establish the most efficient settings for a given set of discovery conditions. our results largely concur with the empirical design of the thousand genomes project, though they furnish some additional refinement.  the optimization principles reported here dramatically simplify the design process and should be broadly useful as rare-variant projects become both more important and routine in the future. 