• We may not know …
  • how to measure (quality, shape, taste)
  • when to measure (development, season, kinetics)
  • what to measure (molecular pathways)
• We are interested in many things
  • individuals expensive
  • multiple measurements cheap (high throughput)
• causal relationships among traits
  • pleiotropy vs close linkage
  • indirect effect of QTL via other trait
  • untangle genetic & environment correlation

• same phenotype in multiple environments
  • does genetic response depend on environment?
  • comparing 2 (or more) environments
  • trends across a cline or gradient
• multiple phenotypes
  • does one trait affect others (directly)?
  • leveraging similar biological function
  • correlation and causalty models

• Time
  • development, growth, aging, senescence
  • physiology, reaction to stimuli
  • evolution, selection
• Space
  • geography, altitude, longitude, height, gravity
  • soil structure, plant density
  • seed & pollen (plant part) dispersal
• Stress
  • physiology, nutrients, light, heat, water
  • weeds, herbivores, pollinators
  • disease, pests, parasites, harvesting

• use multiple phenotypes to improve QTL detection
• do traits share QTL (pleiotropy)?
• causal relationships among traits?
• effects of QTL across time

• Arabidopsis thaliana Ler x Cvi
  • 92 NIL; 2525 seedlings
  • 162 RILs; 2132 (RIL1) or 2325 (RIL2) seedlings

• genotypes: 102 (NIL) or 234 (RILs) markers on 5 chr

• root tip angles every 2 min

Moore, Johnson, Kwak, Livny, Broman, Spalding (2013)

Moore, Johnson, Kwak, Livny, Broman, Spalding (2013)

• \(y_1\) & \(y_2\) are pleiotropic
  • both depend on \(q_1\)
  • cor(\(y_1,y_2|q_1\)) = 0
• \(y_2\) & \(y_3\) are pleiotropic
  • both depend on \(q_2\)
  • cor(\(y_2,y_3|q_2\)) = 0
• \(y_1\) & \(y_3\) have linked QTLs
  • cor(\(q_1,q_2\)) via linkage
  • cor(\(y_1,y_3\)) indirect

one trait: \[ y = q \beta + e, \qquad e \sim \text{N}(0, \sigma^2) \]

\[ \text{LOD} = (n/2) \log_{10} ( \text{RSS}_0 / \text{RSS}_1 ) \]

multiple traits:

\[ Y = Q \boldsymbol{\beta} + E, \qquad E \sim \text{MVN}(0, \Sigma) \]

\[ \text{LOD} = (n/2) \log_{10} ( {|\hat{\Sigma}_0|} / { |\hat{\Sigma}_1|}) \]

• consider two traits and one chromosome
• assume each affected by a single QTL
• 1-D scan for single QTL assumed to affect both
• 2-D scan over chromosome
  • separate axes for each trait
  • or profile each trait across other trait
• significance?
  • parametric bootstrap using fitted single-QTL model
  • stratified (within QTL genotype) permutation test

• allow 2 QTL per trait
  
• is 1 enough?
• or are there 2 QTL?

profile LOD of each trait with respect to the other

• All … phenomena are linked together, and the problem … is how close is the degree of association. Karl Pearson (1911)
• direct influence of one condition on another … [through] all connecting paths of influence … among the variables in a system with … causal relations. Sewall Wright (1921)
• Causality is not mystical or metaphysical. It can be understood in terms of simple processes … in a friendly mathematical language, ready for computer analysis. Judea Pearl (2000)

• \(q_j\) = genotype (QTL)
• \(r_k\) = RNA expression
• \(y\) = physiology

QTLs: \(r_j = \mu_{qj} + e_j\)
linkage: \(\texttt{cor}(q_1,q_2)\)
causal: \(y = r_2 + e_2\)
reactive: \(r_4 = a + by + e_4\)

\[\texttt{pr}(Q,R,Y)= \texttt{pr}(q_1,q_2) \texttt{pr}(y_1|q_1) \texttt{pr}(r_2|q_2) \texttt{pr}(r_3|q_2) \texttt{pr}(y|r_2) \texttt{pr}(r_4|y)\]

• 500 mice F2: B6 x BTBR obese
• diabetes & obesity
• insulin resistance
• 130+ mRNA expression hot spot
• Nfatc2 as key driver

Keller et al. Yandell, Kendziorski, Attie (2016) PLoS Gen

is gene mRNA causal (red) or reactive (blue) to other mRNA?