# Radiation from relativistic shocks with turbulent magnetic fields

K.-I. Nishikawa  J. Niemiec  M. Medvedev  B. Zhang  P. Hardee  Å. Nordlund  J.
Frederiksen  Y. Mizuno  H. Sol  M. Pohl  D. H. Hartmann  M. Oka  G. J. Fishman
National Space Science and Technology Center, Huntsville, AL 35805, USA
Institute of Nuclear Physics PAN, ul. Radzikowskiego 152, 31-342 Kraków,
Poland  Department of Physics and Astronomy, University of Kansas, KS 66045,
USA  Department of Physics, University of Nevada, Las Vegas, NV 89154, USA
Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, AL
35487, USA  Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej
30, 2100 Copenhagen Ø, Denmark  LUTH, Observatore de Paris-Meudon, 5 place
Jules Jansen, 92195 Meudon Cedex, France  Department of Physics and Astronomy,
Iowa State University, Ames, IA 50011, USA  Department of Physics and
Astronomy, Clemson University, Clemson, SC 29634, USA  1Space Sciences
Laboratory, University of California, Berkeley, California 94720, USA
NASA/MSFC, Huntsville, AL 35805, USA

###### Abstract

Using our new 3-D relativistic electromagnetic particle (REMP) code
parallelized with MPI, we investigated long-term particle acceleration
associated with a relativistic electron-positron jet propagating in an
unmagnetized ambient electron-positron plasma. The simulations were performed
using a much longer simulation system than our previous simulations in order
to investigate the full nonlinear stage of the Weibel instability and its
particle acceleration mechanism. Cold jet electrons are thermalized and
ambient electrons are accelerated in the resulting shocks. Acceleration of
ambient electrons leads to a maximum ambient electron density three times
larger than the original value. Behind the bow shock in the jet shock strong
electromagnetic fields are generated. These fields may lead to time dependent
afterglow emission. We calculated radiation from electrons propagating in a
uniform parallel magnetic field to verify the technique. We also used the new
technique to calculate emission from electrons based on simulations with a
small system. We obtained spectra which are consistent with those generated
from electrons propagating in turbulent magnetic fields with red noise. This
turbulent magnetic field is similar to the magnetic field generated at an
early nonlinear stage of the Weibel instability. A fully developed shock
within a larger system generates a jitter/synchrotron spectrum.

###### keywords:

acceleration of particles, galaxies, jets, gamma rays bursts, magnetic fields,
plasmas, shock waves, radiation

, , , , , , , , , , , , and

##  1 RPIC Simulations

Particle-in-cell (PIC) simulations can shed light on the physical mechanism of
particle acceleration that occurs in the complicated dynamics within
relativistic shocks. Recent PIC simulations of relativistic electron-ion and
electron-positron jets injected into an ambient plasma show that acceleration
occurs within the downstream jet nishi03 ; nishi05 ; Hededal & Nishikawa
(2005); nishi06 ; ram07 ; chang08 ; anat08a ; anat08b ; sironi09m . In
general, these simulations have confirmed that relativistic jets excite the
Weibel instability, which generates current filaments and associated magnetic
fields medv99 , and accelerates electrons Hededal & Nishikawa (2005); nishi06
; ram07 ; chang08 ; anat08a ; anat08b ; sironi09m .

Therefore, the investigation of radiation resulting from accelerated particles
(mainly electrons and positrons) in turbulent magnetic fields is essential for
understanding radiation mechanisms and their observable spectral properties.
In this report we present a new numerical method to obtain spectra from
particles self-consistently traced in our PIC simulations.

Figure 1: The averaged values of electron density (a) and field energy (b)
along the $x$ at $t=3750\omega_{\rm pe}^{-1}$. Fig. 1a shows jet electrons
(red), ambient electrons (blue), and the total electron density (black). Fig.
1b shows electric field energy (red) and magnetic field energy (blue) divided
by the total kinetic energy.

Pair Jets Injected into Unmagnetized Pair Plasmas using a Large System

We have performed simulations using a system with ($L_{\rm x},L_{\rm y},L_{\rm
z})=(4005\Delta,$ $131\Delta,131\Delta)$ ($\Delta=1$: grid size) and a total
of $\sim 1$ billion particles (12 particles$/$cell$/$species for the ambient
plasma) in the active grid zones nishi09 . In the simulations the electron
skin depth, $\lambda_{\rm ce}=c/\omega_{\rm pe}=10.0\Delta$, where
$\omega_{\rm pe}=(4\pi e^{2}n_{\rm e}/m_{\rm e})^{1/2}$ is the electron plasma
frequency and the electron Debye length $\lambda_{\rm e}$ is half of the grid
size. Here the computational domain is six times longer than in our previous
simulations nishi06 ; ram07 . The electron number density of the jet is
$0.676n_{\rm e}$, where $n_{\rm e}$ is the ambient electron density and
$\gamma=15$. The electron/positron thermal velocity of the jet is $v^{\rm
e}_{\rm j,th}=0.014c$, where $c=1$ is the speed of light.

Figure 2: The case with a strong magnetic field ($B_{\rm x}=3.7$) and larger
perpendicular velocity ($v_{\perp 1}=0.1c,v_{\perp 2}=0.12c$). The paths of
two electrons moving helically along the $x-$direction in a homogenous
magnetic field shown in the $x-y$-plane (a). The two electrons radiate a time
dependent electric field. An observer situated at great distance along the
n-vector sees the retarded electric field from the moving electrons at the
rest frame (b). The observed power spectrum at different viewing angles from
the two electrons (c). Frequency is in units of $\omega_{\rm pe}^{-1}$.

Figure 1 shows the averaged (in the $y-z$ plane) electron density and
electromagnetic field energy along the jet at $3750\omega_{\rm pe}^{-1}$. The
resulting profiles of jet (red), ambient (blue), and total (black) electron
density are shown in Fig. 1a. The ambient electrons are accelerated by the jet
electrons and pile up towards the front part of jet. At the earlier time the
ambient plasma density increases linearly behind the jet front. At the later
time the ambient plasma shows a rapid increase to a plateau behind the jet
front, with additional increase to a higher plateau farther behind the jet
front. The jet density remains approximately constant except near the jet
front.

Figure 3: Two-dimensional images in the $x-z$ plane at $y/\Delta=65$ for
$t=450\omega_{\rm pe}^{-1}.$ The colors indicate the x-component of current
density generated by the Weibel instability, with the x- and z-components of
magnetic field represented by arrows (a). Phase space distributions as a
function of $x/\Delta-\gamma v_{\rm x}$ plotted for the jet (red) and ambient
(blue) electrons at the same time.

The Weibel instability remains excited by continuously injected jet particles
and the electromagnetic fields are maintained at a high level, about four
times that seen in a previous, much shorter grid simulation system ($L_{\rm
x}=640\Delta$). At the earlier simulation time a large electromagnetic
structure is generated and accelerates the ambient plasma. As shown in Fig.
1b, at the later simulation time the strong magnetic field extends up to
$x/\Delta=2,000$. These strong fields become very small beyond $x/\Delta=2000$
in the shocked ambient region nishi06 ; ram07 .

Figure 4: Two-dimensional images in the $x-z$ plane at $y/\Delta=65$ for
$t=525\omega_{\rm pe}^{-1}.$ The colors indicate the x-component of current
density generated by the Weibel instability, with the x- and z-components of
magnetic field represented by arrows (a). Phase space distributions as a
function of $x/\Delta-\gamma v_{\rm x}$ plotted for the jet (red) and ambient
(blue) electrons at the same time.

The acceleration of ambient electrons becomes visible when jet electrons pass
about $x/\Delta=500$. The maximum density of accelerated ambient electrons is
attained at $t=1750\omega_{\rm pe}^{-1}$. The maximum density gradually
reaches a plateau as seen in Fig. 1a. The maximum electromagnetic field energy
is located at $x/\Delta=1,700$ as shown in Fig. 1b.

###  1.1 New Numerical Method for Calculating Synchrotron Emission

Let a particle be at position ${\bf{r}_{0}}(t)$ at time $t$ nishi08 ; Hededal
(2005); Hededal & Nordlund (2005). At the same time, we observe the electric
field from the particle from position $\bf{r}$. However, because of the finite
velocity of light, we observe the particle at an earlier position
$\bf{r}_{0}(\rm{t}^{{}^{\prime}})$ where it was at the retarded time
$t^{{}^{\prime}}=t-\delta t^{{}^{\prime}}=t-\bf{R}(\rm{t}^{{}^{\prime}})/c$.
Here $\bf{R}(\rm{t}^{{}^{\prime}})=|\bf{r}-\bf{r}_{0}(\rm{t}^{{}^{\prime}})|$
is the distance from the charge (at the retarded time $t^{{}^{\prime}}$) to
the observer.

After some calculation and simplifying assumptions the total energy $W$
radiated per unit solid angle per unit frequency from a charged particle
moving with instantaneous velocity $\boldsymbol{\beta}$ under acceleration
$\boldsymbol{\dot{\beta}}$ can be expressed as Rybicki and Lightman (1979);
Jackson (1999)

$\displaystyle\frac{d^{2}W}{d\Omega d\omega}$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}cq^{2}}{16\pi^{3}}\left|\int^{\infty}_{-\infty}\frac{\bf{n}\times[(\bf{n}-\boldsymbol{\beta})\times\boldsymbol{\dot{\beta}}]}{(1-\boldsymbol{\beta}\cdot\bf{n})^{2}}e^{i\omega(t^{{}^{\prime}}-\bf{n}\cdot\bf{r}_{0}({\rm
t}^{{}^{\prime}})/{\rm c})}dt^{{}^{\prime}}\right|^{2}$ (1)

Here,
$\bf{n}\equiv\bf{R}(\rm{t}^{{}^{\prime}})/|\bf{R}(\rm{t}^{{}^{\prime}})|$ is a
unit vector that points from the particle’s retarded position towards the
observer.

The observer’s viewing angle is set by the choice of $\bf{n}$ ($n_{\rm
x}^{2}+n_{\rm y}^{2}+n_{\rm z}^{2}=1$). The choice of unit vector $\bf{n}$
along the direction of propagation of the jet (hereafter taken to be the
$x$-axis) corresponds to head-on emission. For any other choice of $\bf{n}$
(e.g., $\theta_{\gamma}=1/\gamma$), off-axis emission is seen by the observer.

In order to calculate radiation from relativistic jets propagating along the
$x$ direction nishi08  we consider a test case which includes a parallel
magnetic field ($B_{\rm x}$), and jet velocity of $v_{\rm j1,2}=0.99c$. Two
electrons are injected with different perpendicular velocities ($v_{\perp
1}=0.1c,v_{\perp 2}=0.12c$). A maximum Lorenz factor of
$\gamma_{\max}=\\{(1-(v_{\rm j2}^{2}+v_{\perp 2}^{2})/c^{2}\\}^{-1/2}=13.48$
is calculated with the larger perpendicular velocity.

Figure 2 shows electron trajectories in the $x-y$ plane (red: $v_{\perp
2}=0.12c$, blue: $v_{\perp 1}=0.1c$) (a: left panel), the radiation (retarded)
electric field (b: middle panel), and spectra (right panel) for the case
$B_{\rm x}=3.70$. The two electrons are propagating left to right with
gyration in the $y-z$ plane (not shown). The gyroradius is about $0.44\Delta$
for the electron with the larger perpendicular velocity. The seven curves show
the power spectrum at viewing angles of 0∘ (red), 10∘ (orange), 20∘ (yellow),
30∘ (moss green), 45∘ (green), 70∘ (light blue), and 90∘ (blue). The higher
frequencies become stronger at the $10^{\circ}$ viewing angle. The critical
angle for off-axis radiation $\theta_{\gamma}=180^{\circ}/(\pi\gamma_{\max})$
for this case is 4.25∘. As shown in this panel, the spectrum at a larger
viewing angle ($>20^{\circ}$) has smaller amplitude.

Since the jet plasma has a large velocity $x$-component in the simulation
frame, the radiation from the particles (electrons and positrons) is strongly
beamed along the $x$-axis (jitter radiation) Medvedev (2000, 2006).

Equations 6.30a and 6.30b show that the radiation with the viewing angle
$\alpha=0$ disappears (see Fig. 6.5 in the textbook of Rybicki and Lightman
Rybicki and Lightman (1979)). However, based on two other textbooks, radiation
at the viewing angle $0^{\circ}$ should not vanish beke66 ; land80 . This
aspect is shown in Fig. 2c, and at the higher frequency the amplitude at the
viewing angle $10^{\circ}$ is stronger than that with viewing angle
$0^{\circ}$.

###  1.2 The Standard Synchrotron Radiation Model

A synchrotron shock model is widely adopted to describe the radiation
mechanism in the external shock thought to be responsible for observed broad-
band GRB afterglows Zhang & Meszaros (2004); Piran (2005a, b); Zhang (2007);
Nakar (2007). Associated with this model are three major assumptions that are
adopted in almost all current GRB afterglow models. Firstly, electrons are
assumed to be “Fermi” accelerated at the relativistic shocks and to have a
power-law distribution with a power-law index $p$ upon acceleration, i.e.
$N(E_{\rm e})dE_{\rm e}\propto E^{-p}dE_{\rm e}$. This is consistent with
recent PIC simulations of the shock formation and particle acceleration
anat08b  and also some Monte Carlo models Achterberg et al. (2001); Ellison &
Double (2002); Lemoine & Pelletier (2003), but see Niemiec, & Ostrowski
(2006); Niemiec, Ostrowski, & Pohl (2006). Secondly, a fraction $\epsilon_{\rm
e}$ (generally taken to be $\leq 1$) of the total electrons associated with
ISM baryons are accelerated, and the total electron energy is a fraction
$\epsilon_{\rm e}$ of the total internal energy in the shocked region.
Thirdly, the strength of the magnetic fields in the shocked region is unknown,
but its energy density ($B^{2}/8\pi$) is assumed to be a fraction
$\epsilon_{B}$ of the internal energy. These assumed “micro-physics”
parameters, $p,\epsilon_{\rm e}$ and $\epsilon_{\rm B}$, whose values are
obtained from spectral fits Panaitescu, A., Kumar (2001); Yost et al. (2003)
reflect a lack of knowledge of the underlying microphysics Waxman (2006).

The typical observed emission frequency from an electron with (comoving)
energy $\gamma_{\rm e}m_{\rm e}c^{2}$ in a frame with a bulk Lorentz factor
$\Gamma$ is $\nu=\Gamma\gamma_{\rm e}^{2}(eB/2\pi m_{\rm e}c)$. Three critical
frequencies are defined by three characteristic electron energies. These are
$\nu_{\rm m}$ (the injection frequency), $\nu_{\rm c}$ (the cooling
frequency), and $\nu_{\rm M}$ (the maximum synchrotron frequency). In our
simulations of GRB afterglows, there is one additional relevant frequency,
$\nu_{\rm a}$, due to synchrotron self-absorption at lower frequencies
Meszaros, Rees, & Wijer (1998); Sari, Piran, & Narayan (1998); Zhang (2007);
Nakar (2007).

The general agreement between the blast wave dynamics and the direct
measurements of the fireball size argue for the validity of this model’s
dynamics Zhang (2007); Nakar (2007). The shock is most likely collisionless,
i.e. mediated by plasma instabilities Waxman (2006). The electromagnetic
instabilities mediating the afterglow shock are expected to generate magnetic
fields. Afterglow radiation was therefore predicted to result from synchrotron
emission of shock accelerated electrons Meszaros & Rees (1997). The observed
spectrum of afterglow radiation is indeed remarkably consistent with
synchrotron emission of electrons accelerated to a power-law distribution,
providing support for the standard afterglow model based on synchrotron
emission of shock accelerated electrons Piran (1999, 2000, 2005a); Zhang &
Meszaros (2004); Meszaros (2002, 2006); Zhang (2007); Nakar (2007).

In order to determine the luminosity and spectrum of synchrotron radiation,
the strength of the magnetic field ($\epsilon_{\rm B}$) and the energy
distribution of the electrons ($p$) must be determined. Due to the lack of a
first principles theory of collisionless shocks, a purely phenomenological
approach to the model of afterglow radiation was ascribed without
investigating in detail the processes responsible for particle acceleration
and magnetic field generation Waxman (2006). Rather, one simply assumes that a
fraction $\epsilon_{\rm B}$ of the post-shock thermal energy density is
carried by the magnetic field, that a fraction $\epsilon_{\rm e}$ is carried
by electrons, and that the energy distribution of the electrons is a power-
law, $d\log n_{\rm e}/d\log\varepsilon=p$ (above some minimum energy
$\varepsilon_{0}$ which is determined by $\epsilon_{\rm e}$ and $p$),
$\epsilon_{\rm B}$, $\epsilon_{\rm e}$ and $p$ are treated as free parameters,
determined by observations. It is important to clarify here that the
constraints implied on these parameters by the observations are independent of
any assumptions regarding the nature of the afterglow shock and the processes
responsible for particle acceleration or magnetic field generation. Any model
should satisfy these observational constraints.

The properties of synchrotron (or “jitter”) emission from relativistic shocks
will be determined by the magnetic field strength and structure and the
electron energy distribution behind the shock. The characteristics of jitter
radiation may be important to understanding the complex time evolution and/or
spectral structure in gamma-ray bursts Preece et al. (1998). For example,
jitter radiation has been proposed as a means to explain GRB spectra below the
peak frequency that are harder than the “line of death” spectral index
associated with synchrotron emission Medvedev (2000, 2006), i.e., the observed
spectral power scales as $F_{\nu}\propto\nu^{2/3}$, whereas synchrotron
spectra are $F_{\nu}\propto\nu^{1/3}$ or softer Medvedev (2006). Thus, it is
essential to calculate radiation production by tracing electrons (positrons)
in self-consistently treated small-scale electromagnetic fields.

###  1.3 Calculating Synchrotron and Jitter Emission from Electron
Trajectories in Self-consistently Generated Magnetic Fields

In order to obtain the spectrum of synchrotron (jitter) emission, we consider
an ensemble of electrons selected in the region where the Weibel instability
has fully grown and electrons are accelerated in the generated magnetic
fields. In order to validate our numerical method we performed simulations
using a small system with ($L_{\rm x},L_{\rm y},L_{\rm
z})=(645\Delta,131\Delta,131\Delta)$ ($\Delta=1$: grid size) and a total of
$\sim 0.5$ billion particles (12 particles$/$cell$/$species for the ambient
plasma) in the active grid zones nishi06 . First we performed simulations
without calculating radiation up to $t=450\omega_{\rm pe}^{-1}$. The jet front
is located around about $x/\Delta=480$. We selected 12,150 electrons for each
jet and ambient electrons randomly. Recently, a similar calculation has been
carried out for the radiation from accelerated electrons in laser-wakefield
acceleration Martins et al. (2009) and in shocks Sironi & Spitkovsky (2009).

Figure 3 shows (a) the current filaments generated by the Weibel instability
and (b) the phase space of $x/\Delta-\gamma V_{\rm x}$ for jet electrons (red)
and ambient electrons (blue) at $t=450\omega_{\rm pe}^{-1}$.

Figure 5: Spectra obtained from jet and ambient electrons for the two viewing
angles. Spectra with jet electrons are shown in red ($0^{\circ}$) and orange
($5^{\circ}$). Spectra from ambient electrons show the lowest levels by blue
($0^{\circ}$) and light blue ($5^{\circ}$).

Figure 4 shows (a) the $x$-component of current density generated by the
Weibel instability and (b) the phase space of jet electrons and ambient
electrons after $t_{\rm s}=75\omega_{\rm pe}^{-1}$ (at $t=525\omega_{\rm
pe}^{-1}$).

Figure 6: The case with a weak magnetic field ($B_{\rm x}=0.37$) and small
perpendicular velocity ($v_{\perp 1}=0.01c,v_{\perp 2}=0.012c$). The paths of
two electrons moving helically along the $x-$direction in a homogenous
magnetic field shown in the $x-y$-plane (a). The two electrons radiate a time
dependent electric field. An observer situated at great distance along the
n-vector sees the retarded electric field from the moving electrons (b). The
observed power spectrum at different viewing angles from the two electrons
(c). Frequency is in units of $\omega_{\rm pe}^{-1}$.

We calculated the emission from 12,150 electrons during the sampling time
$t_{\rm s}=t_{\rm 2}-t_{\rm 1}=75\omega_{\rm pe}^{-1}$ with Nyquist frequency
$\omega_{\rm N}=1/2\Delta t=200\omega_{\rm pe}$ where $\Delta
t=0.005\omega_{\rm pe}^{-1}$ is the simulation time step and the frequency
resolution $\Delta\omega=1/t_{\rm s}=0.0133\omega_{\rm pe}$.

The spectra shown in Fig. 5 are obtained for emission from jet electrons and
ambient electrons separately. In this case the spectra are calculated for
head-on radiation ($0^{\circ}$) and $5^{\circ}$. The radiation from jet
electrons show Bremsstrahlung-like spectra as a red line ($0^{\circ}$) and
orange line ($5^{\circ}$) Hededal (2005). The spectra with jet electrons are
different from the spectra shown in Fig. 2c. Since the magnetic fields
generated by the Weibel instability are rather weak and the jet electrons are
not much accelerated, the trajectories of jet electrons are almost straight
with only a slight bent.

We compare these spectra with our known spectra obtained from two (jet)
electrons, the case with a parallel magnetic field ($B_{\rm x}=0.37$), and jet
velocity of $v_{\rm j1,2}=0.99c$. Two electrons are injected with different
perpendicular velocities ($v_{\perp 1}=0.01c,v_{\perp 2}=0.012c$). A maximum
Lorenz factor of $\gamma_{\max}=\\{(1-(v_{\rm j2}^{2}+v_{\perp
2}^{2})/c^{2}\\}^{-1/2}=7.114$ accompanies the larger perpendicular velocity.
The critical angle for off-axis radiation
$\theta_{\gamma}=180^{\circ}/(\pi\gamma_{\max})$ for this case is 8.05∘.

Comparing the spectra with Figs. 5 and 6c we find similarities. The lower
frequencies have flat spectra and the higher frequencies decrease
monotonically. The slope in Fig. 5 is less steep than that in Fig. 6c. This is
due to the fact that the spread of Lorenz factors of jet electrons is larger
and the average Lorenz factor is larger as well. Furthermore, even the
magnetic field strength is not so large, however the slope of the spectra
seems to be consistent with Fig. 7.16 (left) with the turbulent magnetic field
with the red noise ($\mu=-3$) in Hededal’s Ph. D. thesis Hededal (2005). We
obtained several different parameters with jet electrons and ambient magnetic
field. However, the strength of the magnetic fields generated by the Weibel
instability is small, therefore the spectra for these cases are very similar
to Fig. 5. As shown in Fig. 7.12 in Hededal’s Ph. D. thesis Hededal (2005),
the trajectories of jet electrons have to be chaotic to produce a jitter-like
spectrum Fig. 7.22 Hededal (2005).

As shown in Fig. 1b, the magnetic field energy in the region $x/\Delta<500$ is
small ($\epsilon_{\rm B}<0.07$), therefore, as expected, the spectra look like
those emitted from electrons propagating in a turbulent magnetic field with
red noise (see also Figs. 3a and 4a).

###  1.4 Discussions

Emission obtained with the method described above is obtained self-
consistently, and automatically accounts for magnetic field structures on
small scales responsible for jitter emission. By performing such calculations
for simulations with different parameters, we can investigate and compare the
different regimes of jitter- and synchrotron-type emission Medvedev (2000,
2006). The feasibility of this approach has already been demonstrated Hededal
(2005); Hededal & Nordlund (2005), and its implementation is straightforward.
Thus, we should be able to address the low frequency GRB spectral index
violation of the synchrotron spectrum line of death Medvedev (2006).

Medvedev and Spitkovsky recently showed that electrons may cool efficiently at
or near the shock jump and are capable of emitting a large fraction of the
shock energy Medvedev & Spitkovsky (2009). Such shocks are well-resolved in
existing PIC simulations; therefore, the microscopic structure can be studied
in detail. Since most of the emission in such shocks would originate from the
vicinity of the shock, the spectral power of the emitted radiation can be
directly obtained from finite-length simulations and compared with
observational data.

As shown in Fig. 1, behind the trailing shock the electrons are accelerated
and strong magnetic fields are generated. Therefore, this region seems to
produce the emission that is observed by satellites. We will calculate more
spectra based on our RPIC simulations and compare in detail with Fermi data.

###  1.5 Achknowledgments

This work is supported by NSF-AST-0506719, AST-0506666, AST-0908040,
AST-0908010, NASA-NNG05GK73G, NNX07AJ88G, NNX08AG83G, NNX08AL39G, and
NNX09AD16G. JN was supported by MNiSW research projects 1 P03D 003 29 and N
N203 393034, and The Foundation for Polish Science through the HOMING program,
which is supported through the EEA Financial Mechanism.Simulations were
performed at the Columbia facility at the NASA Advanced Supercomputing (NAS).
and IBM p690 (Copper) at the National Center for Supercomputing Applications
(NCSA) which is supported by the NSF. Part of this work was done while K.-I.
N. was visiting the Niels Bohr Institute. Support from the Danish Natural
Science Research Council is gratefully acknowledged. This report was finalized
during the program “Particle Acceleration in Astrophysical Plasmas” at the
Kavli Institute for Theoretical Physics which is supported by the National
Science Foundation under Grant No. PHY05-51164.

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# From Genotype to Phenotype: polygenic prediction of complex human traits

Timothy G. Raben  Michigan State University, East Lansing, Michigan, 48824
Louis Lello  Michigan State University, East Lansing, Michigan, 48824  Genomic
Prediction, North Brunswick, New Jersey, 08902  Erik Widen  Michigan State
University, East Lansing, Michigan, 48824  Stephen D.H. Hsu  Michigan State
University, East Lansing, Michigan, 48824  Genomic Prediction, North
Brunswick, New Jersey, 08902

###### Abstract

Decoding the genome confers the capability to predict characteristics of the
organism (phenotype) from DNA (genotype). We describe the present status and
future prospects of genomic prediction of complex traits in humans. Some
highly heritable complex phenotypes such as height and other quantitative
traits can already be predicted with reasonable accuracy from DNA alone. For
many diseases, including important common conditions such as coronary artery
disease, breast cancer, type I and II diabetes, individuals with outlier
polygenic scores (e.g., top few percent) have been shown to have 5 or even 10
times higher risk than average. Several psychiatric conditions such as
schizophrenia and autism also fall into this category. We discuss related
topics such as the genetic architecture of complex traits, sibling validation
of polygenic scores, and applications to adult health, in vitro fertilization
(embryo selection), and genetic engineering.

A version of this article was prepared for Genomic Prediction of Complex
Traits, Springer Nature book series Methods in Molecular Biology.

Keywords: genomics, complex trait prediction, PRS, in vitro fertilization,
genetic engineering

##  1 Introduction

I, on the other hand, knew nothing, except … physics and mathematics and an
ability to turn my hand to new things. — Francis Crick

The challenge of decoding the genome has loomed large over biology since the
time of Watson and Crick. Initially, decoding referred to the relationship
between DNA and specific proteins or molecular mechanisms, but the ultimate
goal is to deduce the relationship between DNA and phenotype — the character
of the organism itself. How does Nature encode the traits of the organism in
DNA? In this review we describe recent advances toward this goal, which have
resulted from the application of machine learning (ML) to large genomic data
sets. Genomic prediction is the real decoding of the genome: the creation of
mathematical models which map genotypes to complex traits.

It is a peculiarity of ML and artificial intelligence (AI) applied to complex
systems that these methods can often “solve” a problem without explicating, in
a manner that humans can absorb, the intricate mechanisms that lie
intermediate between input and output. For example, AlphaGo [1] achieved
superhuman mastery of an ancient game that had been under serious study for
thousands of years. Yet nowhere in the resulting neural network with millions
of connection strengths is there a human-comprehensible guide to Go strategy
or game dynamics. Similarly, genomic prediction has produced mathematical
functions which predict quantitative human traits with surprising accuracy —
e.g., height, bone density, and cholesterol or lipoprotein A levels in blood
(see Table 1); using typically thousands of genetic variants as input (see
next section for details) — but without explicitly revealing the role of these
variants in actual biochemical mechanisms. Characterizing these mechanisms —
which are involved in phenomena such as bone growth, lipid metabolism,
hormonal regulation, protein interactions — will be a project which takes much
longer to complete.

Phenotype | Correlation | # active SNPs
---|---|---
Height | $0.639_{(0.017)}$ | $22,000_{(3000)}$
Heel bone density | $0.449_{(0.015)}$ | $15,000_{(4000)}$
BMI | $0.346_{(0.0009}$ | $22,000_{(2000)}$
Educational attainment | $0.272_{(0.022)}$ | $17,000_{(7000)}$
Apolipoprotein A | $0.417_{(0.006)}$ | $15,000_{(2,000)}$
Apolipoprotein B | $0.38_{(0.01)}$ | $9,000_{(2,000)}$
Cholesterol | $0.310_{(0.007)}$ | $10,000_{(3,000)}$
Direct bilirubin | $0.51_{(0.01)}$ | $4,000_{(4,000)}$
HDL cholesterol | $0.46_{(0.01)}$ | $17,000_{(2,000)}$
Lipoprotein A | $0.757_{(0.008)}$ | $3,000_{(1,000)}$
Platelet count | $0.45_{(0.01)}$ | $15,000_{(800)}$
Total bilirubin | $0.56_{(0.01)}$ | $5,000_{(3,000)}$
Total protein | $0.32_{(0.01)}$ | $15,000_{(1,000)}$
Triglycerides | $0.348_{(0.008)}$ | $11,000_{(4,000)}$
Table 1: Examples of quantitative trait prediction. The last 10 traits listed
are all obtained from standard blood test measurements (terminology from UK
Biobank data fields). Uncertainties given in parenthesis are the standard
deviation obtained from validation of 5 different training runs, each of which
produce a slightly different predictor. Predictors were trained with data and
methods analogous to [2].

If recent trends persist, in particular the continued growth of large genotype | phenotype data sets, we will likely have good genomic predictors for a host of human traits within the next decade. Here good can mean capturing most of the a priori estimated heritability of the trait. There are already many disease risk predictors which fall somewhat short of this standard but nevertheless have important practical utility in medicine (e.g., for early screening and diagnosis), as we will discuss below. We can roughly estimate how predictor performance increases as a function of training data size (see figure 1). The estimates suggest that progress is data limited: algorithms and computational resources are not the bottleneck.

In the following sections we will discuss (1) Current status of human genomic
prediction: examples of quantitative trait prediction and prediction of
disease risk, out of sample validation of predictors, sibling validation, (2)
Methods, Genetic Architecture, and Theory, and (3) Applications to In Vitro
Fertilization and Gene Editing. The final section will discuss some (near
term) future projections.

Figure 1:  Prediction quality measured by AUC for three conditions
(hypertension, hypothyroidism, type 2 diabetes) as a function of training
sample size $N$. Evidence is strong that further increases in sample size will
lead to improvement in accuracy. Reproduced from [2].

##  2 Present Status: 2020

Technological advances have reduced the (2020) cost of whole genome sequencing
to under $1k and the cost of SNP array genotyping to roughly $20 [3]. In this
section we will give an overview of results obtained from training on data
sets obtained using SNP arrays. For quantitative traits sample sizes were in
excess of 400k individuals. For disease risk training (polygenic risk scores,
or PRS) typical sample sizes were tens of thousands of cases and at least as
many controls, typically individuals late in life for whom medical records are
available. See [2] for specific details.

###  2.1 Quantitative traits: height, bone, and blood

In 2012 one of the authors [4] estimated that training on a few hundred
thousand SNP genotypes using L1 penalization (see next section for details)
would capture most of the common SNP heritability for height. In 2017, with
the release of the UK Biobank data set of 500k genomes, this prediction came
true. Accurate genomic prediction of adult height, with standard deviation of
$\sim 3$ cm, was demonstrated in [5], and has been subsequently replicated by
other groups [6, 7], as well as in studies of siblings. See figures 2 and 6
for a demonstration of the prediction accuracy.

In 2020 the GIANT collaboration, in a GWAS of roughly 4 million individuals,
identified $\sim$10k height SNPs at genome-wide statistical significance. The
variance accounted for by the L1 predictor and by the predictor produced from
the 2020 GIANT GWAS are roughly equal to the previously estimated narrow sense
heritability accounted for by common SNPs [8, 9].

Genomic predictors which capture a significant fraction of heritability exist
for other quantitative traits, including bone density, educational attainment
/ cognitive abilities, and a number of blood measurements such as platelet
count and lipoprotein A levels, see table 1.

Figure 2:  Height prediction in males and females not used in predictor
training. Predicted height computed from $\sim$20k SNPs is shown on vertical
axis, and actual height on horizontal axis. In a typical bin the standard
deviation of the distribution of actual heights relative to prediction is
$\sim 3$cm. While it is possible to find individuals whose height deviates
from prediction by significantly more than a few cm, most of the population
density is concentrated close to the prediction line. Correlation between
predicted and actual (z-scored) height is $\sim 0.65$. Reproduced from [5].

###  2.2 Disease Risks: Polygenic Risk Scores (PRS)

Polygenic risk predictors for dozens of important disease conditions,
including, e.g., diabetes, breast cancer, coronary artery disease,
hypertension, schizophrenia, autism, and many more, have been published and
validated by many research groups [2, 10, 11, 12].

We can roughly characterize the performance of these polygenic risk predictors
as follows: individuals with very high PRS will typically have an incidence
rate which is many times higher than the population average. For example, in
[2] we found that for atrial fibrillation, 99th percentile PRS implies
$\sim$10 times higher likelihood of case status. Similarly, low PRS indicates
below average risk for the condition: we can identify individuals whose risk
is an order of magnitude lower than in the general population.

Identification of outliers is possible even though the standard performance
metric AUC (Area Under ROC Curve) value is modest: e.g., AUC $\sim 0.6$. This
is because the absolute risk as function of PRS is highly nonlinear: outlier
(e.g., 99th percentile) risk can be very high even if risk for individuals
near the middle of the PRS distribution varies only modestly — see figure 3,
which illustrates risk differentiation for the example phenotypes breast
cancer and hypothyroidism. We report below on further explicit results.

Figure 3:  Incidence of breast cancer and hypothyroidism as a function of
percentile polygenic risk score (PRS). At high PRS the likelihood of incidence
increases nonlinearly, and at low PRS the likelihood decreases nonlinearly.
Red curve is theoretical, modeling case and control populations with normal
distributions shifted in mean PRS. Blue data points are calculated using
individuals (not used in training) binned by PRS. Reproduced from [2].

There is already significant interest in the application of PRS in a clinical
setting, for example to identify high risk individuals who might receive early
screening or preventative care [13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 22, 23,
24]. As a concrete example, women with high PRS scores for breast cancer can
be offered early screening: already standard of care for those with BRCA risk
variants [25, 26]. However, BRCA mutations affect no more than a few women per
thousand in the general population [27, 28, 29]. Importantly, the number of
(BRCA negative) women who are at high risk for breast cancer due to polygenic
effects is an order of magnitude larger than the population of BRCA carriers
[10, 2, 30, 31, 32, 33, 34]. From this one example it is clear that
significant medical, public health, and cost benefits could result from PRS
(e.g. [35]). It is well known that patients with atherosclerotic diseases,
coronary artery disease (CAD), and lung diseases can benefit from early
intervention [36, 37, 38]. In many instances where early treatment can be
beneficial, stratification by age, gender, and ethnicity show an exacerbation
of poor outcomes [39]. Precision genetics is already used in identification of
candidates for early intervention, and will become widespread in the near
future (cf. Myriad’s riskScore test and other examples [34, 33]). In figure 4,
we illustrate the predicted risk of breast cancer and coronary artery disease
as function of age for high, medium and low risk groups, respectively.

Figure 4:  Breast cancer and coronary artery disease (CAD) risk progression
with age and polygenic score (PRS). Outlier individuals with unusually high
(top 4 percent) PRS are much more likely to be diagnosed with the condition
than those with typical (40th to 60th percentile) PRS. Low PRS is associated
with reduced incidence of the condition. Similar results are available for 20
or more common disease conditions, and have obvious utility for early
screening, diagnosis, and prevention. PRS obtained from [10] and scored on a
population within the UK Biobank [40].

It should be stressed that here we focus on _purely genomic_ risk scores and
correlations. That is, we are focused on relative genetic risk from SNP
information alone. These results can be easily combined with information from
other biomarkers (e.g., blood test results) or health-related indicators such
as age, sex, BMI, blood pressure, etc. to obtain even stronger risk
stratification [2, 10, 11, 15].

PRS with similar utility for risk outlier identification have been developed
for psychiatric conditions such as autism, schizophrenia [41, 42].

###  2.3 Sibling Validation

There are now many validations of polygenic prediction in the scientific
literature, conducted using groups of people born on different continents and
in different decades than the original populations used in training [43, 44].
Here we discuss results showing that predictors can differentiate between
siblings (which one has heart disease? is taller?), despite similarity in
childhood environments and genotype. The predictors work almost as well in
pairwise sibling comparisons as in comparisons between randomly selected
strangers.

We tested a variety of polygenic predictors using tens of thousands of genetic
siblings for whom we have SNP genotypes, health status, and phenotype
information in late adulthood. Siblings have typically experienced similar
environments during childhood, and exhibit negligible population
stratification relative to each other. Therefore, the ability to predict
differences in disease risk or complex trait values between siblings is a
strong test of genomic prediction in humans. We compare validation results
obtained using non-sibling subjects to those obtained among siblings and find
that typically most of the predictive power persists in within-family designs.
Given 1 sibling with normal-range PRS score (less than 84th percentile) and 1
sibling with high PRS score (top few percentiles), the predictors identify the
affected sibling about 70-90 percent of the time across a variety of disease
conditions, including breast cancer, heart attack, diabetes, etc. For height,
the predictor correctly identifies the taller sibling roughly 80 percent of
the time when the (male) height difference is 2 inches or more. Some disease
prediction results are illustrated in figure 5 while a sibling validation of
height prediction can be seen in figure 6.

Figure 5:  Predictors tested on random (non-sibling) pairs and affected
sibling pairs with a single case, for conditions coronary artery disease
(CAD), type 1 diabetes (T1D), hypothyroidism, and hypertension. One individual
in each pair is high risk (i.e., has a high polygenic risk score) and the
other is normal risk ($\text{PRS}<+1\text{SD}$). The difference in PRS
z-scores is given on the horizontal axis. The individual classified as high
risk by the predictor is likely to be the one to exhibit the condition,
increasingly so as the z-score difference becomes large. Quality of prediction
is very similar between pairs of random individuals and sibling pairs, despite
the siblings having experienced more similar childhood environments and
sharing more alleles in common. Error bands include uncertainty due to limited
numbers of individuals in each z-score bin. Reproduced from [45].  Figure 6:
Probability that polygenic predictor correctly identifies the taller
individual (vertical axis) for pairs of random individuals and pairs of
siblings. Horizontal axis shows absolute difference in z-scored height between
the individuals in each pair. Quality of prediction is very similar between
pairs of random individuals and sibling pairs, despite the siblings having
experienced more similar childhood environments and sharing more alleles in
common. Error bands include uncertainty due to limited numbers of individuals
in each z-score bin. Reproduced from [45].

##  3 Methods, Genetic Architecture, and Theoretical Considerations

###  3.1 Sparse learning, L1 penalization, phase transition behavior

Sparse learning algorithms have been successfully applied to construct genomic
predictors [46, 5, 47, 48]. These algorithms incorporate a prior that SNPs
which materially affect the trait are only a small fraction of the (typically
of order million or more) candidate SNPs. In other words, the algorithms favor
parsimony in the construction of models for genetic risk. This prior has been
confirmed by many studies: most of the population variance for even the most
polygenic traits (e.g., human height) is captured by at most tens of thousands
of SNPs [49]. Although 10k is a large number, it is small compared to the
millions of candidate common SNPs (i.e., polymorphisms found in at least $\sim
1$ percent of the population) in each person’s DNA. Hence, the assumption of
sparsity has strong empirical support [50].

Our lab has successfully used a sparse learning technique called L1 penalized
regression, also known as LASSO or compressed sensing. Below we elaborate on
our methodology in detail.

In analyses performed to date, LASSO, and penalized regression in general,
performs as well or better than other training algorithms [51] — such as
logistic regression, support vector machines, linear mixed-models, random
forests, Bayesian regression [52, 53, 54, 55, 56, 57, 58, 59] — for trait
prediction. Studies comparing sparse learning against other methods, including
neural networks, have not found a consistent advantage; it is fair to say that
currently sparse learning methods are comparable to or better than the
alternatives [60, 61, 62].

Our standard process for building a sparse predictor was designed to optimize
performance _within_ an specific ancestry group, as self-reported or according
to PCA clustering. We elaborate more on cross-ancestral studies later in this
section while for this method, both training and validation is performed
within a specified ancestry.

Predictor training

In order to avoid difficulties arising from population structure, training is
performed in a homogeneous population with similar ancestry. Standard tools
using, e.g., principal components analysis, allow efficient categorization by
ancestry. The set of candidate SNPs is typically either the full set of SNPs
directly measured by the genotyping array, or a larger set obtained by
imputation. Basic quality control is performed to avoid using extremely rare
variants and poorly genotyped participants. The weights ${\beta_{j}}$ are
chosen by minimizing the objective function

$O=\tfrac{1}{2}|\vec{y}-X\vec{\beta}|^{2}+\lambda\sum_{j}|\beta_{j}|$

for the vector of phenotypes $y$ and matrix $X$ of SNP genotypes for each
sample. $\lambda$ is a hyperparameter of the model which tunes the level of
sparsity imposed. The predictors are trained using $k$-fold cross-validation:
a small subset of data is withheld from training and used for model selection;
the entire process is repeated $k$ times with a different subset withheld
every time.

Score/Validation

Each trained predictor is scored on its corresponding validation subset
withheld from its $k$-fold training. We typically use standard prediction
metrics such as AUC and explained variance to determine optimal parameter
settings and to select top predictors from each cross-validation fold.

Evaluation

Once the optimal predictors have been selected they can be evaluated in a
number of ways. Typically we use evaluation data sets composed of (1)
individuals of similar ancestry to the training set, but not used in training,
(2) individuals of adjacent ancestry not used in training (e.g., Eastern or
Southern Europeans, adjacent to British / North-West Europeans), (3)
individuals of generally similar ancestry but collected in entirely different
cohorts, sometimes from another continent, (4) distant ancestry groups (e.g.,
non-Europeans of a specific ancestry such as East Asian or African), and most
prominently (5) _siblings_ of generally European ancestry (see [45] for a full
paper on sibling evaluation). We again compute standard prediction metrics
such as AUC, but also more relevant measures such as the absolute probability
(rate of incidence) of the condition in outlier subgroups such as top few
percent PRS score, as illustrated in figure 3.

Predictors built with L1 penalization typically have between 100 and 20k
active SNPs, depending on phenotype, distributed over many chromosomes — see
figure 7 for the example of height. The predictor performance varies with the
trait but is always strongly dependent on the sample size available for
training. Empirically, we find an approximate sample size dependence
$\sim\tfrac{N}{N+b}$ with the asymptotic behavior determined by the
(linear/narrow-sense) heritability of the trait in question and by the number
of candidate SNPs. In practice, there is typically a very steep gain in
prediction power as the sample sizes grow from small to moderate, say from 1k
to 50k samples. It is then followed by a region of less dramatic but steady
performance gains until it eventually flattens out for extremely large data
sets. Figure 1 shows how predictor performance improves with increasing sample
sizes for hypertension, hyperthyroidism and type 2 diabetes.

Figure 7:  Locations of $\sim 20$k SNPs activated in height predictor on the
genome, with individual chromosomes indicated. Vertical axis is effect size
$\beta$ of minor allele. Positive $\beta$ value indicates a SNP for which the
minor allele is associated with increased height. Reproduced from [5].

We have shown, e.g., for height and bone-heel mineral density, that the UKB
with its $\sim$ 400k individuals of European ancestry is currently in the late
second region. For many traits there is still substantial predictive power to
be gained from increased sample size, as seen in figure 1.

Sparse learning algorithms using L1 penalization have been demonstrated to be
highly effective. Our original interest in these methods arose because, almost
uniquely, there are strong mathematical results characterizing their behavior.
For example, celebrated theorems (largely unknown to the computational
genomics community) provide performance guarantees for compressed sensing [47]
as a function of signal/noise ratio and training data size. Further, it is
known that these algorithms display a kind of universal phase transition
behavior as the data size is varied. In [46] we showed that matrices of human
SNP genotypes are good compressed sensors, and exhibit the celebrated Donoho-
Tanner phase transition behavior previously found for large classes of random
matrices. These results were further verified in [63, 5].

###  3.2 Prediction across ancestry groups, causal variants

PRS training has so far overwhelmingly been conducted in populations of
homogeneous ancestry, typically of European descent, for which large databanks
first became available. This is because population stratification (patterns of
correlation within the genome, which differ by ancestry) introduces special
difficulties in statistical learning (e.g. [64]). Consequently, the majority
of predictors have been trained on and work best in European populations.
There are a few exceptions in which GWAS works well in diverse populations,
e.g. [65], but performance of complex PRS fall off quickly as a function of
genetic distance [66, 67]. The implications of this skewed focus are serious
as the majority of the world population, including minorities within countries
of predominantly European descent, are left out from these new advancements in
health care [68]. It is thus an urgent priority to correct this situation: (1)
by building predictors using cohorts from other ancestries (e.g., East Asian
or West African) as well as (2) developing techniques that can modify or adapt
predictors trained in one ancestry group so that they work well in another,
perhaps distant, ancestry group.

Cross-ancestral study and training of predictors provide a unique opportunity
to explore the genetic architecture for common diseases, a research area in
which important basic questions still remain. While PRS and GWAS identify
candidate SNPs which are statistically associated with increased (or
decreased) risk, they cannot determine which SNPs have a causal effect on
individual biology. Because SNPs often occur in correlated clusters in the
genomes of a given population, there is always some ambiguity concerning
whether a specific SNP is causal. These correlation patterns vary across
populations.

Current predictors utilize SNPs which are merely tags (i.e., correlated in
state) for the actual causal SNPs (or other structure). The quality of the tag
may be much weaker in a distant population, causing the predictor to perform
much worse. This is a problem to be solved.

By utilizing different patterns of correlation, we may be able to zero in on
actual causal SNPs — they are likely to be consistently detected in predictor
training across multiple distant populations. In other words, we can turn the
problem described above into a tool for detecting candidate causal SNPs. See
for instance [69] for early results for CAD and [70] for a current overview of
how cross-population studies have furthered the understanding of Alzheimer’s
Disease.

###  3.3 Coding regions, Pleiotropy

Using the SNPs activated in existing PRS, we can begin to investigate the
genetic architecture of common disease risk [71]. There are many detailed
results concerning specific conditions, but two general points should be
emphasized:

  1. 1.

Much of the genetic risk identified in polygenic predictors is controlled by
variants outside genic (protein coding) regions, and not accessible through
exome sequencing. This supports the notion that DNA information storage
extends beyond specific genes.

  2. 2.

The DNA regions used in disease risk predictors so far constructed seem to be
largely disjoint, suggesting that most genetic disease risks are largely
uncorrelated. It seems possible in theory for an individual to be a low-risk
outlier in all conditions simultaneously.

The space of genetic variation is high dimensional, and extends far beyond
individual (protein coding) genes. Intuitions about strong pleiotropy are
likely wrong — they were developed before we knew anything about real genetic
architectures. There seem to be many causal variants that could, in principle,
be independently modified and evidence to date suggests that large portions of
genetic variance affecting different human traits and disease risks are
independent.

In the final section below, we make some rough estimates concerning the total
space of heritable individual differences (including both quantitative traits
and disease risks) for humans, assuming approximate independence.

###  3.4 Linearity / Additivity

There is significant empirical evidence now that linear predictive models can
capture much (nearly all?) of the estimated common SNP heritability of many
traits. It may come as a surprise that genetic effects can be approximately
additive, given the apparent complexity of biological systems.

Nonlinear genetic effects certainly exist and are likely realized in every
organism. However, quantitative differences between individuals within a
species may be largely due to independent linear effects of specific genetic
variants. As reflected in Fisher’s Fundamental Theorem of Natural Selection
[4], linear effects are the most readily evolvable in response to selection,
whereas nonlinear “gadgets” (i.e., mechanisms which depend sensitively on
multiple genetic switches) are more likely to be fragile to small changes.
Evolutionary adaptations requiring significant changes to nonlinear gadgets
are improbable and therefore require exponentially more time than simple
adjustment of frequencies of alleles of linear effect. One might say that to
first approximation, Biology = linear combinations of nonlinear gadgets, and
most of the variation between individuals in a species is due to the (linear)
way gadgets are combined, rather than in the realization of different gadgets
in different individuals.

##  4 In Vitro Fertilization and Genetic Engineering

Today millions of babies are produced through In Vitro Fertilization (IVF). In
most developed countries roughly 3-5 percent of all births are through IVF,
and in Denmark the fraction is about 10 percent [72]. But when the technology
was first introduced with the birth of Louise Brown in 1978, the pioneering
scientists had to overcome significant resistance.

> Wikipedia: …During these controversial early years of IVF, Fishel and his
> colleagues received extensive opposition from critics both outside of and
> within the medical and scientific communities, including a civil writ for
> murder. Fishel has since stated that "the whole establishment was outraged"
> by their early work and that people thought that he was "potentially a mad
> scientist". [73]

In the past, parents with more viable embryos than they intended to use made a
selection based on very little information — typically nothing more than the
appearance or morphology of each blastocyst. With modern technology it has
become common to genotype embryos before selection, in order to detect
potential genetic issues such as trisomy 21 (Down Syndrome). Parents who are
carriers of a single gene variant linked to a Mendelian condition can use
genetic screening to avoid passing the risk variant on to their child.
Millions of embryos are now genetically tested each year. With polygenic risk
prediction, it is possible now to screen against outlier risk for many common
disease conditions, not just rare single gene conditions. For example, the
overwhelming majority of families with breast cancer history are not carriers
of a BRCA risk variant, but rather have elevated polygenic risk. It is now
possible for these families to select an embryo with average or even below
average breast cancer risk if they so wish.

Beyond IVF embryo selection, the advent of CRISPR and other recent advances in
genetic editing suggest that future technologies will permit germ line editing
of humans — perhaps leading to consequences in human evolution.

Note that for genomic prediction it is enough to identify SNPs which are
correlated to the phenotype. But to achieve the desired effect in editing one
must identify the actual causal genetic variants. This step in the research
program is highly nontrivial and may take longer to accomplish than the
development of the molecular editing tools.

Highly polygenic traits imply a very large reservoir of extant variance
already present in the population [4]. It is this extant variance that plant
and animal breeders have used to advance agriculture for thousands of years.
Roughly speaking, if a trait is controlled by $\sim N$ genetic variants, an
increase in phenotype by one population standard deviation corresponds to
changing $\sim N^{1/2}$ variants from the $(-)$ to $(+)$ state. Thus the
maximum number of standard deviations that can be captured through editing
could be as large as $N^{1/2}$!

The population geneticist James Crow of Wisconsin wrote [74]:

> The most extensive selection experiment, at least the one that has continued
> for the longest time, is the selection for oil and protein content in maize
> (Dudley 2007). These experiments began near the end of the nineteenth
> century and still continue; there are now more than 100 generations of
> selection. Remarkably, selection for high oil content and similarly, but
> less strikingly, selection for high protein, continue to make progress.
> There seems to be no diminishing of selectable variance in the population.
> The effect of selection is enormous: the difference in oil content between
> the high and low selected strains is some 32 times the original standard
> deviation.

To take another example, wild chickens lay eggs at the rate of roughly one per
month. Domesticated chickens have been bred to lay almost one egg per day.
(Those are the eggs we have for breakfast!) Of all the wild chickens in
evolutionary history, probably not a single one produced eggs at the rate of a
modern farm chicken.

The corresponding ethical issues are complex and wide ranging. They deserve
serious attention in what may be a relatively short interval before these
capabilities become a reality in human genetic engineering and widespread
clinical practice. We cannot do them justice here, but they include topics
such as: the power dynamic between population geneticists and studied
populations [75]; personal and communal identity issues [76]; comparisons of
different types of pre-implantation testing [77, 78]; how to develop ethical
guidelines in a fast changing field [79, 80]; disease specific concerns [81];
misinformation and media bias [82]; congenital vs adult-onset testing [83];
non-medical testing and sex selection [84, 85]; provider duties and
obligations [86]; disparities in health care [67]; intersection of legal and
religious concerns with genetics [87, 88]; evolution of the limits of gene
editing [89]; ethical statements and disclosures of those using CRISPR [90];
patent competition and human application of CRISPR [91]; animal welfare [92];
and the extensive and intertwining history of genetics and eugenics [93, 94,
95, 96].

Each society will decide for itself where to draw the line on human genetic
engineering, but we can expect a diversity of perspectives. Almost certainly,
some countries will allow genetic engineering, thereby opening the door for
global elites who can afford to travel for access to reproductive technology.
As with most technologies, such as IVF, the rich and powerful will be the
first beneficiaries. Eventually, though, it is possible that many countries
will not only legalize human genetic engineering, but even make it a
(voluntary) part of their national healthcare systems. The alternative would
be inequality of a kind never before experienced in human history.

##  5 The Future

We conclude with some predictions concerning future progress, and some
unifying theoretical remarks related to high dimensionality and its role in
genomic prediction.

Perhaps the easiest prediction to make is that there are still significant
gains to be made from simply increasing the training data size. Already for
some common disease conditions we can identify outliers (e.g., few percent of
the population) who have well over 50 percent probability to have the disease
by late adulthood. This level of prediction will become available for many
more conditions, and the size of the identifiable high risk population will
increase considerably. There will be important consequences for early
screening, diagnosis, and prevention in medical care. Health insurance may
also be transformed: individuals who know their risk profiles have an
asymmetric information advantage over insurers. Perhaps we will see a day when
no insurer will price a policy without DNA analysis. Or, perhaps strong
genomic prediction will force societies into a single payer healthcare
structure, in which all risks are pooled.

Next we list some startling developments that can already be anticipated and
only await sufficient training data to be realized.

  1. 1.

Face Recognition: AI algorithms use of order 100 features to identify human
faces (e.g., distance between the eyes, or from nose to mouth, etc.). From
identical twins, we know that these features — each a complex quantitative
trait — is itself highly heritable. With enough training data, already
conveniently extracted by face recognition algorithms from ordinary photos, we
expect that facial features will be predictable from DNA and hence faces
themselves can be reconstructed from DNA alone. This will likely have
applications in forensic science (e.g., to solve crimes) as well as in IVF
(parents will have an idea of what their child will look like, at difference
ages, from an embryo genotype). It will also intersect with the ethical use of
facial recognition technology and the protection of civil liberties [97, 98,
99].

  2. 2.

Cognitive and Personality traits: Substantial progress has been made in the
prediction of cognitive ability [44, 100]. Actual cognitive score correlates
roughly 0.3 to 0.4 with predicted score. Quality of prediction appears to be
entirely data limited, and we expect these correlations to increase
considerably before the regime of diminishing returns is reached. Personality
traits such as conscientiousness, extraversion, or agreeableness are known to
be highly heritable, and we expect them to be predictable from genotype once
sufficient phenotype data become available [101]. In general, behavioral
traits are more greatly influenced by environmental factors than other
phenotypes, exacerbating the above mentioned ethical concerns.

  3. 3.

Longevity: As we mentioned in our discussion of pleiotropy, genetic disease
risks seem to be largely (although not completely) independent. This suggests
that (at least theoretically), individuals could exist that have low risk
across all common disease conditions that impact life expectancy. In the
future we may be able to identify outliers in longevity, and understand better
the limits to human life span.

We can begin to formulate a “grand unified theory” of human individual
differences, using an information theoretic approach and what we already know
about the genetic architecture and dimensionality of the space of human
variation [102]. Some orders of magnitude:

  * •

$\sim$10M common SNP differences between two individuals.

  * •

$\sim$10k SNPs may control most of the variance for a typical complex trait.

We can take the number of common (e.g., MAF $>0.01$) SNP differences that is
typical between two individuals as an estimate of the amount of genomic
information that determines the (heritable) individual differences among
humans. In principle, there could be $\sim$1k (a few thousand) largely
independent complex traits with little pleiotropy between them. These might
include a hundred common disease or health risks, hundreds of cosmetic traits,
including facial and body morphology parameters, dozens of psychometric
variables, including personality traits, etc.

Clearly, individual differences are well accommodated by a $\sim$1k
dimensional phenotype space embedded in a $\sim$10M dimensional space of
genetic variants.

Of course, it is an unrealistic idealization for the traits to be entirely
independent in genetic architecture. We expect to find pleiotropy at some
level: a subset of genetic variants will affect more than one trait. But these
estimates suggest that a significant part of the genetic variance of each
trait can be predicted using existing linear models, and potentially modified
(e.g., via editing) independently of the other traits. This is simply a
consequence of high dimensionality.

It seems certain that genomic prediction will have significant impacts on
health care through better screening, diagnosis, and treatment of almost all
important disease conditions. As we have emphasized the main limiting factor
is the availability of sufficiently large data sets with good phenotype
information across populations of diverse ancestry. Computational and
algorithmic methods are not (at least at the moment) the main constraint on
progress. Perhaps some day soon we will have good predictors for almost all
heritable individual differences in the human species. The dream of decoding
the human genome is within our reach and important societal choices concerning
how to apply the results are upon us.

#### Competing Interests

Stephen Hsu a shareholder of Genomic Prediction, Inc. (GP), and serves on its
Board of Directors. Louis Lello is an employee and shareholder of GP. Timothy
Raben and Erik Widen have no commercial interests relevant to this research.

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