DNA forensic genealogy solved the Golden Gate Killer case

The alleged murderer of UnitedHealth CEO Brian Thompson was recently caught, identified by an observant customer and employee at a Pennsylvania McDonald’s. While he was still on the run, I was confident he would eventually be caught once I heard they had obtained a DNA sample from saliva on a bottle or cup. There has been a recent revolution in DNA forensics, which has led to many cold cases being solved (Wikipedia). In these cases, a sample of DNA from the alleged perpetrator was left at the scene of the crime, but police did not have a person to match the DNA with.

Over the past 20 years, we have developed science fiction level technology to sequence DNA quickly, cheaply, and from a small sample. It is now possible to sequence a whole human genome for close to $100 in a single day. One only needs to remember the sequencing of the first human genome over a period of roughly 10 years (1990-2001) for a cost of approximately $5 billion (present day dollars) to appreciate the insane progress. In the not-too-distant future, we all will have our genomes sequenced.

Intuitively, people understand that the DNA sequence of an individual is unique (excepting identical twins), and that DNA sequences of close relatives will be more similar than the DNA sequences of two strangers, but can we be more precise? For each of the 23 pairs of chromosomes in nearly every cell in the human body, you inherit one chromosome (single molecule of DNA) from Mom (maternal) and one chromosome from Dad (paternal). Is the chromosome you inherit from Mom (say chromosome 1) exactly the same as the chromosome in all of Mom’s cells? The answer is no, because recombination occurs during meiosis (special cell division that produces the gametes) so that the chromosome 1 found in a particular gamete (egg) is a mix of your Mother’s maternal and paternal chromosome 1. During recombination, segments of DNA are exchanged between the two homologous chromosomes in a pair resulting in recombinant chromosomes.

I previously mentioned the concept of identity-by-descent (IBD) which refers to shared, completely identical DNA segments between two or more individuals inherited from a common ancestor. Two siblings share 50% of their DNA via identity-by-descent. Each sibling inherits a chromosome 1 from Mom. Meiotic recombination described in the previous paragraph mixes the chromosome so that half is from Mom’s Mom and half is from Mom’s Dad. This mixing is random so that one sibling may get a particular segment from Mom’s Mom, whereas the other sibling has a 50% chance of getting the identical segment from Mom’s Mom but also a 50% chance of getting the segment from Mom’s Dad. Thus, for siblings, 50% of their maternal chromosome 1 will be shared from a common ancestor (i.e. Mom). The same applies for their paternal chromosome 1.

Without going into detail, for each succeeding generation the IBD sharing decreases by a factor of 4, and so first cousins share $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8} = 12.5\%$ DNA, and 2nd cousins $\frac{1}{8} \times \frac{1}{4} = \frac{1}{32} = 3.125\%$. Briefly, first cousins related through their fathers share 25% of their paternal chromosome 1; their maternal chromosome 1 is 0% IBD since their Moms are not related. Meiotic recombination will produce various first chromosomes that are 50% paternal and 50% maternal which will be passed down to their kids. Any previously shared segment in the paternal chromosome 1 between the cousins has only a 1/2 chance of being passed down in each cousin resulting in IBD sharing that decreases by $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ between the second cousins compared to the first cousins. 

Thus, it is straightforward to calculate the amount of DNA shared between various types of relatives as shown in the Table below:

Relationship	Average % DNA Shared	Range (cM)*
Identical Twin	100%	N/A
Parent / Child	50%	~3719
Full Sibling	50%	2826 - 4537
Grandparent / Grandchild	25%	1264 - 2529
Aunt / Uncle	25%	1264 - 2529
Niece / Nephew	25%	1264 - 2529
Half Sibling	25%	1264 - 2529
1st Cousin	12.5%	298 - 1710
2nd Cousin	3.13%	149 - 446
3rd Cousin	0.78%	0 - 164
4th Cousin	0.20%	0 - 60

*1 centimorgan (cM) is approximately 1 million base pairs (1 Mb) in humans.

Beyond 5th cousins, the amount of shared DNA becomes very small and difficult to distinguish from random background sharing. In other words, short identical stretches of DNA may not be inherited from a recent common ancestor, but rather reflect chance similarities.

How can DNA genealogy be used to track down a criminal, i.e. pick out a suspect based on a DNA sample as described in the first paragraph? One can take advantage of large commercial genealogy (ancestry) databases to identify distant relatives of the suspect, e.g 5th cousin or closer. How big of a database do we need, and what is the probability of obtaining a hit?

Yaniv Erlich and colleagues provided approximate answers to these questions in the article "Identity inference of genomic data using long-range familial searches" which describes the use of consumer genomic databases for identifying individuals through distant familial relatives. Combining population genetics theory and computer simulations, the authors reached the following conclusions:

• About 60% of searches for individuals of European descent in a database of 1.28 million will find a third-cousin or closer match.
• With a database covering just 2% of a target population, it is theoretically possible to find a third-cousin match for nearly anyone in that population.
• For U.S. individuals of European descent, a database of approximately 3 million could provide a third-cousin match for over 99% of the population.

Thus, a database of 1 million samples is large enough for this purpose, and many such ancestry databases exist that the public can search for matches (relatives).

What is the next step after a hit has been found in an ancestry database? Using the relative as a starting point, authorities can construct a family tree with the goal of eventually adding the suspect to the tree, i.e. build the tree out to the 3rd (or 4th or 5th) cousins of the hit. Then basic demographic information (e.g. man or woman, age, etc.) can be used to pinpoint the suspect within the tree. 

In an investigation, the family tree is often constructed by hand, but in the paper, the authors took advantage of "86 million profiles from publicly-available online data shared by genealogy enthusiasts. After extensive cleaning and validation, we obtained population-scale family trees, including a single pedigree of 13 million individuals."

From these population scale family trees, each hit (in simulations) produced on average a list of ~850 individuals who possessed the appropriate degree of DNA sharing. Then, "localizing the target to within 160 km (100 miles) will exclude 57% of the candidates on average. Next, availability of the target’s age to within ±5 years will exclude 91% of the remaining candidates. Finally, inference of the biological sex of the target will halve the list to just around 16 to 17 individuals, a search space that is small enough for manual inspection.” Thus based on this analysis it should be possible from location, age, and sex, to narrow down the possible suspects to a small handful of people that can be investigated one by one.

In theory, one sequences the DNA sample left behind by an unknown suspect (i.e. hair, saliva, semen, etc.), and uploads the sequence to large commercial ancestry databases containing a million or more entries. One can expect an approximate 3rd cousin hit. A family tree is constructed from this match (either by hand or in an automated fashion taking advantage of online genealogy information) that is likely to encompass the suspect. Demographic information can cull this tree to narrow in on the suspect.

How well does this procedure work in practice? It turns out quite well, and one example is provided by the saga of the Golden Gate Killer, one of the most celebrated cold cases. 

The Golden State Killer is a moniker for a serial offender who committed a series of crimes in California between 1974 and 1986, including burglaries, rapes, and murders. Earlier in his criminal career (1974–1975), he was involved in over 100 burglaries in Visalia, often described as the "Visalia Ransacker." In the mid-1970s, the offender was responsible for over 50 sexual assaults in the Sacramento area and other parts of Northern California. Later, in Southern California, he committed at least 13 murders between 1979 and 1986 and was known as the "Original Night Stalker." By 2001, the authorities were able to link the various cases to the same culprit via DNA who became known as the Golden State Killer (GSK). Unfortunately, there were no new leads and so the case went cold.

However, in 2016 investigators made a renewed effort taking advantage of the DNA samples from GSK and advances in DNA forensic genealogy. They uploaded the DNA sequence to several consumer genetic databases (~1-2 million customers each), and obtained 3rd cousin hits. In Feb. 2018, they uploaded the sequence to the MyHeritage website and did even better with a 2nd cousin hit.

Starting with this second cousin, they built a family tree using the consumer genealogy site Ancestry.com. They then obtained DNA from a member of the family tree to exclude certain people in one portion of the tree (i.e. person and his close relatives were not an exact match with GSK DNA). Only six male cousins remained as possible fits. An FBI search of California driver’s license records showed that only one of those six men had blue eyes consistent with the profile (from DNA sequence it is possible to predict blue or brown eyes with reasonable accuracy which will be the topic of a future blog post).

After 10 days of surveillance that included police enlisting the help of a garbage truck driver to snatch DNA-bearing items from his trash can, Joseph James DeAngelo was arrested on April 24, 2018 and charged with multiple counts of murder. They were able to exactly match DeAngelo's DNA to the DNA of the Golden State Killer which provided conclusive proof. He is currently serving 26 life sentences with no possibility of parole. More information on the GSK case can be found in the following LA Times article, and an excellent video by Veritasium (see below).

TL;DR: Authorities can sequence a tiny sample of DNA from a criminal suspect, and then upload that sequence to ancestry databases. For large databases (> 1 million individuals), they can roughly expect a 3rd cousin match or so. A family tree constructed from the match may include the suspect. Basic demographic information can help authorities prune the tree leaving a small number of remaining candidates from which they can obtain DNA to match with the original sample. If the authorities did indeed have a DNA sample from the killer of Brian Thompson, then eventually his DNA would have led to his capture even without the timely aid of an observant public.

Heritability

In a previous post I described “The Fundamental Law of Genetics,” which states that phenotype (observed trait) = genotype (genetic factors) + environment (environmental factors). The quantitative version of the law is $V_P = V_G + V_E$ in which the phenotypic variance in a population equals genotypic variance (phenotypic variation caused by differences in genotype)  plus environmental variance (phenotypic variance caused by differences in environment). Then I cautioned that the equation is a simplification that omits genotype-by-environment interactions.

In a more recent post I described an alternative version of this law as $z$ (phenotypic value) $= G$ (gentoypic value) $+ E$ (residual or environmental value). A phenotypic value is a measurement of a trait like your height, and the genotypic value is the portion attributed to genotype with the remainder being attributed to environment. This value equation can be converted into the variance equation by taking the variance of both sides and then throwing out the covariance term between G and E.

Intuitively, any trait (such as blood pressure) depends on genetic factors (alleles) that you inherit from your parents as well as environmental factors (such as diet and exercise). But how much is genetics and how much is environment? The concept of heritability attempts to estimate the fraction of a trait that is due to genetics (your genotype).

More quantitatively, heritability is the  fraction of total phenotypic variance ($V_P$) that can be explained by genotypic variance ($V_G$):

$ \begin{equation} H^2 = \frac{V_G}{V_P} = \frac{V_G}{V_G + V_E} \tag{1} \end{equation} $

where $H^2$ refers to broad-sense heritability. My students asked whether they needed to take the square root to obtain $H$, but no, $H^2$ not $H$ is defined to be heritability (good question as always).

Estimating heritability is easier said than done, rife with assumptions and caveats. Further, there is a wide gap between the technical definition of heritability, and the common-sense understanding among the general public (e.g. the apple doesn't fall far from the tree) who tend to overestimate heredity and underestimate rearing. Many of these issues are beyond the scope of this post, and here I will focus on one simple approach to estimating heritability by comparing the phenotypes of parent and child.

In the early 1900s, Francis Galton gathered data highlighting the similarities between parents and children. I have reproduced Figure 8.2 from Hartl and Clark which shows a scatter plot of the weight of male offspring on the y-axis with the weight of their father on the x-axis. The fact that this trait is inherited to a certain degree is reflected in the linear relationship between the parental trait value and the offspring trait value, so that the data points can be fit with a line (i.e. linear regression, which Galton helped to invent). More importantly, this line has a positive slope so that higher offspring values correspond to higher parental values.

Approximate reproduction of Figure 8.2 from Hartl and Clark (2007). Mean weight of male offspring of flour beetle (y-axis) plotted against weight of father (x-axis).  

On the one hand, if the heritability was very low for the trait, then the data points would be randomly scattered and the best line to fit the points would be horizontal of slope 0. On the other hand, if the heritability was close to 1, then the children would have the same value of the trait as their parent (in the simplest case), and the points would lie along a line of slope 1. 

The converse is not true; points lying along a line of slope 1 do not imply that the heritability is 1. There could be environmental components which are identical for both parents and children, e.g. both grew up in wealthy households. Environmental covariance refers to related environments (e.g. for parents and offspring) leading to related phenotypes. However, if we assume that the environmental covariance is zero, we can calculate heritability from the linear regression slope.

We can simplify matters by defining the "midparent" which is the average phenotype of the Mother and Father, and plot the offspring phenotypic values (y-axis) against the midparent phenotypic values (x-axis). From statistics, the resulting regression line that best fits these points will have slope $b = \frac{\operatorname{Cov}(x,y)}{\operatorname{Var}(x)}$.

The phenotypic covariance between child ($y$) and midparent ($x$) is a measure of trait resemblance, i.e. how correlated are the two phenotypes (correlation is scaled covariance). As stated above, one can write the phenotypic value $z$ as the sum of the genotypic value $G$ (contribution of genetic factors) and the environmental value $E$ (contribution of environmental factors): $z = G + E$.

In the general case for any two individuals $x$ and $y$, the phenotypic covariance can written as: $\operatorname{Cov}(x,y)  = \operatorname{Cov}(G_x + E_x, G_y + E_y) = \operatorname{Cov} (G_x,G_y) + \operatorname{Cov}(G_x,E_y) + \operatorname{Cov}(E_x,G_y) + \operatorname{Cov}(E_x,E_y)$ where $G_x$ represents the genotypic values and $E_x$ the environmental values for individual $x$. We will assume that the genotype-environment covariances ($\operatorname{Cov}(G,E)$ terms) are 0 so that we can focus on the remaining two terms.

The term $\operatorname{Cov}(E_x,E_y)$ describes the relatedness of the environments of the two. If $x$ and $y$ were random individuals, then we would expect this covariance to be 0, but as mentioned above, for parent and child many environmental conditions are likely to be similar (“I’m raising my kid just like I was raised”) so that the covariance would not be zero for many traits. Assuming this covariance is 0 is the biggest weakness of calculating heritability from parent-offspring regression, but for now we will assume that it is 0 to continue with the blog post using the derivations outlined in Coop (2020), Hartl and Clark (2007), and Lynch and Walsh (1998).

The remaining term is the covariance of genotypic values $\operatorname{Cov} (G_x,G_y)$. As discussed in the previous post on dominance and epistasis, each genotypic value can be broken up into additive, dominance and epistatic terms: $G_x = G_{xA} + G_{xD} + G_{xS}$. We can then write $\operatorname{Cov} (G_x,G_y) = \operatorname{Cov}(G_{xA},G_{yA}) + Cov(G_{xD},G_{yD}) + \operatorname{Cov}(G_{xS},G_{yS}) +$ cross terms that are 0. For parent and offspring, $\operatorname{Cov}(G_{xD},G_{yD}) = 0$, and most of the epistasis terms are also 0 or presumed to be small (see Table 7.2 in Chapter 7, Lynch and Walsh for details). This leaves us with $\operatorname{Cov} (G_x,G_y) \sim \operatorname{Cov}(G_{xA},G_{yA})$. As a reminder, the additive term is from the linear model of summing the contributions of individual alleles that influence a trait, whereas dominance and epistasis represent nonadditive genetic contributions.

In close relatives, one can calculate the covariance of the additive genotypic values by following the descent of alleles to the respective individuals. Given a locus, an offspring inherits one allele from one parent and one allele from the other parent. The additive covariance between parent and child is 0 for the allele not inherited from the parent (correlation = 0), and is $\frac{V_A}{2}$ for the inherited allele, in which $V_A$ is the additive genotypic variance for the locus with the $\frac{1}{2}$ arising from that a locus consists of two alleles. Thus, for the locus we have $\operatorname{Cov}(G_{xA},G_{yA}) = 0 + \frac{V_A}{2} = \frac{V_A}{2}$. Similar reasoning applies to calculating the additive genotypic covariance between offspring and midparent.

The above is an example of the concept of identity-by-descent (IBD) which refers to shared, completely identical DNA segments between two or more individuals inherited from a common ancestor. A parent shares 50% of DNA with a child through IBD, and so 50% of the offspring's alleles are identical to the parent's through direct descent.

As described above, we can write the slope $b$ of the midparent-offspring regression as the covariance divided by the variance of the midparent which is $Var(x) = \frac{V_P}{2}$ because the midparent is the phenotypic average of the two parents. This leads to the following equation:

$ \begin{equation} b = \frac{\operatorname{Cov}(x,y)}{\operatorname{Var}(x)} \sim \frac{\operatorname{Cov}(G_{xA},G_{yA})}{\operatorname{Var}(x)} = \frac{V_A}{V_P} = h^2 \tag{2} \end{equation} $

The expression $h^2 = \frac{V_A}{V_P}$ is referred to as narrow-sense heritability which consists only of the additive component of the genotypic variance (omitting the dominance and epistatic terms) divided by the total phenotypic variance, and hence it is always less than or equal to broad-sense heritability. However, both theory and empirical data indicate that $h^2$ is close to $H^2$ (Benjamin, et al. 2024). 

Thus, we can calculate the narrow-sense heritability by performing linear regression of offspring phenotypic values against parental phenotypic values for a trait. The main limitation is the assumption that the environmental covariance of parent and offspring is 0 ($\operatorname{Cov}(E_x,E_y) = 0$). If the covariance is non-zero, then the estimate of $h^2$ will be upwardly biased because there is an additional positive term in the numerator of Eq 2 that shouldn't be there. Because of this suspect assumption, researchers in the past have turned to a different method using twins to estimate heritability, the subject of a future post.

TL;DR: There is broad-sense and narrow-sense heritability which describe the fraction of a trait that is attributed to genetic factors. The latter can be estimated by parent-offspring regression, but with the key assumption that environmental covariance is 0. However, it is very unlikely that the parental environment is uncorrelated with the offspring environment. The apple and the tree grow on the same patch of land.