Probabilistic analysis of recessive mutagenesis screen strategies

Abstract

Random mutagenesis screens for recessive phenotypes require three generations of breeding, using either a backcross (BC) or intercross (IC) strategy. Hence, they are more costly and technically demanding than those for dominant phenotypes. Maximizing the return from these screens requires maximizing the number of mutations that are bred to homozyosity in the G₃ generation. Using a probabilistic approach, we compare different designs of screens for recessive phenotypes and the impact each one has on the number of mutations that can be effectively screened. We address the issue of BC versus IC strategies and consider genome-wide, region-specific screens and suppressor screens. We find that optimally designed BC and IC screens allow the screening of, on average, similar numbers of mutations but that interpedigree variation is more pronounced when the IC strategy is employed. By conducting a retrospective analysis of published mutagenesis screens, we show that, depending on the strategy, a threefold difference in the numbers of mutations screened per animal used could be expected. This method allows researchers to contrast, for a range of experimental designs, the cost per mutation screened and to maximize the number of mutations that one can expect to screen in a given experiment.

Appendix: Genome-wide efficiency derivations

Probability of screening a given mutation

We begin by deriving the probability that no homozygotes (for a particular mutation) are born in a GPF. We fix the number of pups per G₂ father, r, but not the number of pups per G₂ female. We assume any pup could have come from any female with equal probability. This also includes the possibility that all pups came from the same female. To denote the genotype of mice at a mutation locus, let “m” and “+” represent the mutant and wild-type alleles (respectively). For the BC,

$$ \eqalign{ &{\rm Pr}\, ({{\hbox{no m/m pups from a GPF with}} \, k{\hbox{ females producing }}r{\hbox{ pups in total}}}) \cr &= {\sum\limits_{i = 0}^k {{\rm Pr} \,{({{\hbox{no m/m pups}}|k{\hbox{ females, }}i{\hbox{ of whom are m}}}/ \,+ )} \ {\rm Pr} \,{( {i{\hbox{ of }}k{\hbox{ females are m}}}/ \,+ )}} } \cr &= {\sum\limits_{i = 0}^k {\left(\matrix{ {k} \cr {i}}\right)} {\left({1 \over 2}\right)}^{i} {\left({1 \over 2}\right)}^{k - i} {\rm Pr} \,({\hbox{pup not m/m}}|k{\hbox{ females, }}i{\hbox{ of whom are m}}/\, +)^{r}} \cr &= {\left( {1 \over 2} \right)}^{k} {\sum\limits_{i = 0}^k {\left(\matrix{{k} \cr {i} }\right)} {\left[ \matrix{ {\rm Pr}\, ({{\hbox{pup not m/m}}}|{{\hbox{mother + / + }}})\,{\rm Pr}\,({ {\hbox{mother +/+ }}|k{\hbox{ females, }}i{\hbox{ are m/ + }}} ) \cr \ + {\rm Pr} \,( {{\hbox{pup not m/m}}}|{{\hbox{mother m/ + }}} ) \,{\rm Pr} \,({{\hbox{mother m/ + }}|k{\hbox{ females, }}i{\hbox{ are m/ + }}} )}\right]}^{r}} \cr &= {\left( {1 \over 2} \right)}^{k} {\sum\limits_{i = 0}^k }{\left(\matrix{{k} \cr {i} }\right)} {\left[ {1 \cdot {\left( {1- {i \over k}} \right)} + {\left( {3 \over 4}\right)} {\left(i \over k \right)} } \right]}^{r} \cr &= {\left( {1 \over 2} \right)}^{k} {\sum\limits_{i = 0}^k }{\left(\matrix{{k} \cr {i} }\right)} {\left[ {1 + {\left( {i \over k} \right)}{\left( {3 \over 4} - 1 \right)}} \right]}^{r} \cr &={\left( {1 \over 2} \right)}^{k} {\left( {1 \over {4k}} \right)}^{r} {\sum\limits_{i = 0}^k } {\left(\matrix{ {k} \cr {i}}\right)} ({4k - i})^{r} \cr &{\rm Pr}\, ({\hbox{at least 1 m/m pup from a box with }}k{\hbox{ females producing }}r{\hbox{ pups in total}} ) \cr &= 1 - {\left( {1 \over 2} \right)}^{k} {\left( {1 \over {4k}} \right)}^{r} {\sum\limits_{i = 0}^k } {\left(\matrix{ {k} \cr {i} }\right)} ({4k - i})^{r} } $$

In the case of the IC, there can be several GPFs per G₁ mating pair (h > 1). Hence,

$$ \eqalign{ &{\rm Pr}\,{\left( \matrix{ {\hbox{at least 1 m/m pup from }}h{\hbox{ GPFs (from the same G}}_{{\hbox{1}}} {\hbox{ pair),}} \aftergroup\hfill \cr {\hbox{each with }}k{\hbox{ females with }}r{\hbox{ pups in total}} \aftergroup\hfill \cr} \right)} \cr &= 1 - {\rm Pr}\,{\left( \matrix{ {\hbox{no m/m pups from }}h{\hbox{ GPFs (from the same G}}_{{\hbox{1}}} {\hbox{ pair), }} \aftergroup\hfill \cr {\hbox{each with }}k{\hbox{ females with }}r{\hbox{ pups in total}} \aftergroup\hfill \cr} \right)} \cr &= 1 - {\rm Pr}\,{\left( {{\hbox{no m/m pups from a given GPF}}} \right)}^{h} \cr &= 1 - {\left[ \matrix{ {\rm Pr}\,( {{\hbox{no m/m pups from a given GPF}}|{\hbox{male is m/ + }}})\,{\rm Pr}\,( {{\hbox{male is m/ + }}} ) \aftergroup\hfill \cr + {\rm Pr}\,({{\hbox{no m/m pups from a given GPF}}|{\hbox{male is + / + }}})\,{\rm Pr}\,( {{\hbox{male is + / + }}}) \aftergroup\hfill \cr} \right]}^{h} \cr &= 1 - {\left[ {{\rm Pr}\,( {{\hbox{no m/m pups from a given GPF}}|{\hbox{male is m/ + }}} ){1 \over 2} + {1 \over 2}} \right]}^{h} \cr &= 1 - {\left[ {{\left( {1 \over 2} \right)}^{k + 1} {\left( {1 \over {4k}} \right)}^{r} {\sum\limits_{i = 0}^k {{\left( {\matrix{ {k} \cr {i} \cr } } \right)}{\left( {4k - i} \right)}^{r} } } + {1 \over 2}} \right]}^{h} } $$

Alternatively, we could fix the number of pups per G₂ female, n, rather than the number of pups per G₂ male. Then for the BC,

$$ \eqalign{ &{\rm Pr}\, ( {{\hbox{at least 1 m/m pup from }}k{\hbox{ females, each with }}n{\hbox{ pups}}} ) \cr &= 1 - {\rm Pr} \,( {{\hbox{no m/m pups from }}k{\hbox{ females, each with }}n{\hbox{ pups}}} ) \cr & = 1 - {\prod\limits_{i = 1}^k }{\rm Pr}\, {\left( {{\hbox{no m/m pups from the }}i^{\rm th} {\hbox{ female}}} \right)} \cr &= 1 - {\rm Pr}\, ( {{\hbox{no m/m pups from a given female}}} )^{k} \cr & = 1 - {\left[ \matrix{ {\rm Pr} \,( {{\hbox{no m/m pups from a given female}}|{\hbox{female is m/ + }}}){\rm Pr} \,( {{\hbox{female is m/ + }}} ) \aftergroup\hfill \cr + {\rm Pr} \,({{\hbox{no m/m pups from a given female}}|{\hbox{female is + / + }}} ){\rm Pr} \,( {{\hbox{female is + / + }}} ) \aftergroup\hfill \cr} \right]}^{k} \cr &= 1 - {\left[ {1 \over 2}{\left( {3 \over 4} \right)}^{n} + {1 \over 2} \right]}^{k} } $$

and for the IC,

$$ \eqalign{& {\rm Pr}\, {\left( \matrix{ {\hbox{at least 1 m/m pup from }}h{\hbox{ GPFs (from the same G}}_{1} {\hbox{ pair),}} \aftergroup\hfill \cr {\hbox{each with }}k{\hbox{ females, each with }}n{\hbox{ pups}} \aftergroup\hfill \cr} \right)} \cr &= 1 - {\rm Pr} \,{\left( \matrix{ {\hbox{no m/m pups from }}h{\hbox{ GPFs (from the same G}}_{1} {\hbox{ pair), }} \aftergroup\hfill \cr {\hbox{each with }}k{\hbox{ females, each with }}n{\hbox{ pups}} \aftergroup\hfill \cr} \right)} \cr &= 1 - {\rm Pr}\, ( {{\hbox{no m/m pups from a given GPF}}} )^{h} \cr &= 1 - {\left[ \matrix{ \Pr ( {{\hbox{no m/m pups from a given GPF}}|{\hbox{male is m/ + }}})\,{\rm Pr} ( {{\hbox{male is m/ + }}} ) \aftergroup\hfill \cr + \Pr ( {{\hbox{no m/m pups from a given GPF}}|{\hbox{male is + / + }}} )\,{\rm Pr} ( {{\hbox{male is + / + }}} ) \aftergroup\hfill \cr} \right]}^{h} \cr & = 1 - {\left[ {\rm Pr} ( {{\hbox{no m/m pups from a given GPF}}|{\hbox{male is m/ + }}} ){1 \over 2} + {1 \over 2} \right]}^{h} \cr &= 1 - {\left[ {{1 \over 2}{\left( {{1 \over 2}{\left( {{3 \over 4}} \right)}^{n} + {1 \over 2}} \right)}^{k} + {1 \over 2}} \right]}^{h} } $$

Let us define P _IC(h,θ) to be the probability of producing at least one G₃ homozygote with the IC, where θ denotes parameters k and n or r (depending on whether we assume that the number of pups per G₂ female is fixed or flexible, respectively). Similarly, define P _BC(1,θ) for the BC. Note here that P _IC(1,θ) = 0.5P _BC(1,θ) under both sets of assumptions.

Mutation screen using the BC

Let X be the number of mutations in the region being screened that are carried by the G₁ founder, and let E(X) = x. Assuming that mutations are segregating independently, then

$$ M_{\rm MBC} (1,X,1,1,\theta) \mathop=\limits^d {\rm Bi} (X,P_{\rm BC} (1,\theta)) \mathop =\limits^d {\rm Bi}(X,2P_{\rm IC}( 1,\theta)), {\rm so} $$

,so

$$\eqalign {Eff_{\rm MBC} (1,x,d,1,\theta) = d{\rm E}( M_{\rm MBC} ( 1,X,1,1,\theta)) &= d{\rm E}({\rm E}( M_{\rm MBC} (1,x,1,1,\theta)|X)) =d{\rm E}(2X{\rm P}_{\rm IC} (1,\theta)) \cr &=2d{\rm P}_{\rm IC} (1,\theta){\rm E}(X)=2dx{\rm P}_{\rm IC}(1,\theta)} $$

Mutation screen using the IC

Let X ₁ and X ₂ be the numbers of mutations carried by the two G₁ founders (in the target region), both with mean x. If we again assume no linkage,

$$ M_{\rm MIC} ( {1,X_{1} + X_{2} ,1,h,\theta }) \mathop =\limits^d{\rm Bi}( {X_{1} + X_{2},P_{\rm IC} ( {h,\theta })}) $$

, then

$$\eqalign{ Eff_{\rm MIC} ({1,x,d,h,\theta}) &= d{\rm E}( {M_{\rm MIC} ( {1,X_{1} + X_{2} ,1,h,\theta} ) )} = d{\rm E}( {\rm E}( {M_{\rm MBC} ( {1,X_{1} + X_{2} ,1,h,\theta } )}|X_{1} + X_{2} )) \cr &= d{\rm E}( ( {X_{1} + X_{2}} ){\rm P}_{\rm IC} ( {h,\theta ))} = 2d{\rm P}_{\rm IC} ( {h,\theta ){\rm E}( {X_{1} + X_{2} } )} = 2dx{\rm P}_{\rm IC} ( {h,\theta )}. } $$

Suppressor screen using the BC²

Now consider screens for recessive suppressors of a recessive phenotype that use the BC. Let M _sBC(1,X,d,1,k,r) be the number of mutations for which at least one G₃ pup is homozygous, and homozygous for the sensitized allele. Results from the above derivations are applicable provided that we count only homozygous mutations in those G₃ that are homozygous for the sensitized allele. Let Z _j be the number of G₃ pups in the jth GPF that are homozygous for the sensitized allele. Note that the Z _j are independent and identically distributed. Then

$$ {M_{\rm sBC} ( {1,X,1,1,k,r})} \mathop = \limits^d {\rm Bi} {\left( {X,1 - \left( {1 \over 2}\right)^{k} {\left( {1 \over {4k}} \right)}^{Z_{j}} {\sum\limits_{i = 0}^k } {\left( \matrix{ {k} \cr {i} } \right)} ( {4k - i} )^{Z_{j} }} \right)}, {\rm where}\, Z_{j} \mathop = \limits^d {\rm Bi}{\left( {r,{1 \over 4}} \right)} $$

So

$$ \eqalign{ Eff_{\rm sBC} ( {1,x,d,1,k,r} ) &= d{\rm E} ({M_{\rm sBC} ( {1,X_{1},1,1,k,r} )}) = d{\rm E}( {\rm E}({M_{\rm MBC} ( {1,X_{1},1,1,k,z_{1}})|Z_{1} = z_{1})}) \cr &= d{\rm E}( {\rm E}({\rm E}( {M_{\rm MBC} ({1,X_{1},1,1,k,z_{1}})|X_{1}})|Z_{1} = z_{1})) \cr &= d{\rm E}{\left( {\rm E}{\left( X_{1} {\left( 1 - {\left( {{1 \over 2}} \right)}^{k} {\left( {{1 \over {4k}}} \right)}^{z_{1} } \sum\limits_{i = 0}^k {\left( \matrix{ {k} \cr {i} \cr } \right)}( {4k - i} )^{z_{1}} \right)} |Z_{1} = z_{1} \right)} \right)} \cr &= d{\rm E}{\left( X_{1} {\left( 1 - {\left( {{1 \over 2}} \right)}^{k} {\left( {{1 \over {4k}}} \right)}^{Z_{1} } \sum\limits_{i = 0}^k {\left( \matrix{ {k} \cr {i} \cr } \right)}( {4k - i} )^{Z_{1}} \right)}\right)} \cr &= d{\rm E}(X_{1}) {\rm E}{\left( 1 - {\left( {{1 \over 2}} \right)}^{k} {\left( {{1 \over {4k}}} \right)}^{Z_{1} } \sum\limits_{i = 0}^k {\left( \matrix{ {k} \cr {i} \cr } \right)}( {4k - i} )^{Z_{1}} \right)} \cr &= dx{\left( 1 - {\left( {{1 \over 2}} \right)}^{k} {\rm E}{\left({\left( {{1 \over {4k}}} \right)}^{Z_{1} } \sum\limits_{i = 0}^k {\left( \matrix{ {k} \cr {i} \cr } \right)}( {4k - i} )^{Z_{1}} \right)} \right)}\cr &= dx{\left( 1 - {\left( {1 \over 2} \right)}^{k} \sum\limits_{j = 0}^r {\left( \matrix{ {r} \cr {j} \cr } \right) {\left( {1 \over 4} \right)}^{j} {\left( {3 \over 4} \right)}^{r - j} {\left( {1 \over {4k}} \right)}^{j} } \sum\limits_{i = 0}^k {\left( \matrix{ {k} \cr {i} \cr } \right)} ({4k - i})^{j} \right)} \cr &= dx{\left( 1 - {\left( {1 \over 2} \right)}^{k} \sum\limits_{j = 0}^r {\left( \matrix{ {r} \cr {j} \cr } \right) {\left( {1 \over 4} \right)}^{j} {\left( {1 \over 4} \right)}^{r - j} 3^{r - j} {\left( {1 \over {4k}} \right)}^{j} } \sum\limits_{i = 0}^k {\left( \matrix{ {k} \cr {i} \cr } \right)} ({4k - i})^{j} \right)} \cr Eff_{\rm sBC} ({1,x,d,1,k,r} ) &= dx{\left( {1 - {\left( {{1 \over 2}} \right)}^{k} {\left( {{3 \over 4}} \right)}^{r} \sum\limits_{j = 0}^r {\left( \matrix{ {r} \cr {j} \cr } \right)} \left( {1 \over {12k}} \right)^{j} \sum\limits_{i = 0}^k \left( \matrix{ {k} \cr {i} \cr } \right)} ( {4k - i})^{j} \right)}}$$

If the sensitized allele is dominant rather than recessive, then we need to count homozygous mutations in those G₃ that carry at least one copy of the sensitized allele. Then

$$ M_{\rm SBC} (1,X_{1},d,1,k,r) {\mathop = \limits^d } {\rm Bi} {\left( X_{1} ,1 - {\left( {1 \over 2} \right)}^{k} {\left( {1 \over {4k}} \right)}^{Y_{j} } \sum\limits_{i = 0}^k \left( \matrix{ {k} \cr {i} \cr } \right) ({4k - i})^{Y_{j}} \right)} , \, {\rm where} \, Y_{j} \mathop = \limits^d {\rm Bi} \left( {r,{3 \over 4}} \right) $$

So

$$ \eqalign{ Eff_{\rm SBC} ({1,x,d,1,k,r}) &=dx{\left( 1 - {\left( {1 \over 2} \right)}^{k} \sum\limits_{j = 0}^r \left( \matrix{ {r} \cr {j} \cr } \right) \left( {3 \over 4} \right)^{j} \left( {1 \over 4} \right)^{r - j} \left( {1 \over {4k}} \right)^{j} \sum\limits_{i = 0}^k \left( \matrix{ {k} \cr {i} \cr } \right) \left( {4k - i} \right)^{j} \right)} \cr & = dx{\left( 1 - \left( {1 \over 2} \right)^{k} \left( {1 \over 4} \right)^{r} \sum\limits_{j = 0}^r \left( {\matrix{ {r} \cr {j} \cr } } \right) {\left( {3 \over {4k}} \right)}^{j} \sum\limits_{i = 0}^k \left( {\matrix{ {k} \cr {i} \cr } } \right) \left( {4k - i} \right)^ {j} \right) }} $$

Suppressor screen using the IC

Using a similar argument as above, we relate results from the BC to the IC.

$$ \eqalign{ &{\rm M}_{\rm sIC} (1,X_{1}+X_{2},1,h,k,r){\mathop = \limits^d } {\rm Bi} {\left( X_{1}+X_{2},1 - {\prod\limits_{j = 1}^h} {\left[ {\left( {1 \over 2} \right)}^{k + 1} {\left( {1 \over {4k}} \right)}^{Z_{j} } {\sum\limits_{i = 0}^k} {\left( \matrix{ {k} \cr {i} \cr } \right)} {\left( {4k - i} \right)}^{Z_{j} } + {1 \over 2} \right]} \right)}, \cr & {\rm where} Z_{j} {\mathop = \limits^d }{\rm Bi} {\left( {r,{1 \over 4}} \right)} \cr & Eff_{\rm sIC} ({1,x,1,h,k,r}) = {\rm E}(M_{\rm sIC} ( {1,X_{1}+X_{2},1,h,k,r} ))= {\rm E} ({\rm E}({M_{\rm sIC} ( {1,X_{1} + X_{2},1,h,k,r} )}|X_{1} X_{2}) ) \cr & \quad= {\rm E}(({X_1+X_2}) M_{\rm sIC} ({1,1,1,h,k,r}))= {\rm E}(X_1+X_2) {\rm E}(M_{\rm sIC} ({1,1,1,h,k,r})) \cr &\quad \quad \quad = 2x{\rm E}({\rm E}( M_{\rm sIC} ( {1,1,1,h,k,r} )| (Z_{1} , \ldots ,Z_{h} ))) \cr & E(M_{\rm sIC} ( {1,1,1,h,k,r} )|({Z_{1} , \ldots ,Z_{h} } ) =( z_{1} , \ldots ,z_{h})) \cr & \quad= 2{\left( 1 - {\prod\limits_{j = 1}^h} {\left[ {\left( {1 \over 2} \right)}^{k + 1} {\left( {1 \over {4k}} \right)}^{z_{j} } {\sum\limits_{i = 0}^k } {\left( \matrix{ {k} \cr {i} \cr } \right)} {\left( {4k - i} \right)}^{z_{j}} + {1 \over 2} \right]} \right)} \cr & Eff_{\rm sIC} ( {1,x,1,h,k,r} ) = 2x {\left( 1 - {\left[ {\left( {1 \over 2} \right)}^{k + 1} {\rm E} {\left[ {\left( 1 \over {4k} \right)}^{Z_{1}} {\sum\limits_{i = 0}^k } {\left( \matrix{ {k} \cr {i}} \right)} {\left( {4k - i} \right)}^{Z_{1}} \right]} + {1 \over 2} \right]}^{h} \right) }\cr & \quad {\hbox{ since the}} \, Z_{j} \, {\hbox {are I.I.D.}} \cr & = 2x {\left( 1 - {\left[ {1 \over 2} + {\left( {1 \over 2} \right)}^{k + 1} {\left( {3 \over 4} \right)}^{r} {\sum\limits_{j = 0}^r } {\left( \matrix{ {r} \cr {j} \cr } \right)} {\left( 1 \over {12k} \right)}^{j} {\sum\limits_{i = 0}^k } {\left( {\matrix{ {k} \cr {i} } } \right)} {\left( {4k - i} \right)}^{j} \right]}^{h} \right)}, \,{\rm and } \cr & Eff_{\rm sIC} ({1,x,d,h,k,r}) = dEff_{\rm sIC} ({1,x,1,h,k,r})} $$

Again, suppose that the sensitized allele is dominant, then

$$ \eqalign{ & {\rm M}_{\rm sIC} ({1,X_1 +X_2,1,h,k,r}) {\mathop = \limits^d } {\rm Bi} {\left( X_{1}+X_{2},1 - {\prod \limits_{j = 1}^h} {\left[ {\left( {1 \over 2} \right)}^{k + 1} {\left( {1 \over {4k}} \right)}^{Y_{j}} {\sum\limits_{i = 0}^k } {\left( \matrix{ {k} \cr {i}} \right)} {\left( {4k - i} \right)}^{Y_{j}} + {1 \over 2} \right]} \right)}, \cr &{\rm where} \, Y_{j} {\mathop = \limits^d} {\rm Bi} {\left( {r,{3 \over 4}} \right)} \cr & Eff_{\rm sIC} ({1,x,1,h,k,r}) = 2x {\left(1 - {\left[ {\left( {1 \over 2} \right)}^{k + 1} {\left[ {\sum\limits_{j = 0}^r } {\left( \matrix{ {r} \cr {j} } \right)} {\left( {3 \over 4} \right)}^{j} {\left( {1 \over 4} \right)}^{r - j} {\left( {1 \over {4k}} \right)}^{j} {\sum\limits_{i = 0}^k } {\left( \matrix{ {k} \cr {i} } \right)} {\left( {4k - i} \right)}^{j} \right]} + {1 \over 2} \right]}^{h} \right) }\cr & \quad = 2x {\left( 1 - {\left[ {1 \over 2} + {\left( {1 \over 2} \right)}^{k + 1} {\left( {1 \over 4} \right)}^{r} {\sum\limits_{j = 0}^r } {\left( \matrix{ {r} \cr {j} } \right)} {\left( {3 \over {4k}} \right)}^{j} {\sum\limits_{i = 0}^k } {\left( {\matrix{ {k} \cr {i} } } \right)} {\left( {4k - i} \right)}^{j} \right]}^{h} \right)} \cr & Eff_{\rm sIC} ({1,x,d,h,k,r}) = 2dx {\left( 1 - {\left[ {1 \over 2} + {\left( {1 \over 2} \right)}^{k + 1} {\left( {1 \over 4} \right)}^{r} {\sum\limits_{j = 0}^r } {\left( \matrix{ {r} \cr {j} } \right)} {\left( {3 \over {4k}} \right)}^{j} {\sum\limits_{i = 0}^k } {\left( \matrix{ {k} \cr {i} }\right)} {\left( {4k - i} \right)}^{j} \right]}^{h} \right)}} $$