prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How is the function beta(r) determined in the derivation?

Read the following text and answer briefly.

Context 1: \section{Introduction}
The Schwarzschild solution plays a key role in teaching about general relativity: It describes the simplest version of a black hole  By Birkhoff's theorem, it more generally describes the gravitational field around any spherical mass distribution, such as the Sun in our own Solar system  As one of two particularly simple, yet physically relevant examples of a non-trivial metric (the other being the FLRW spacetime of an expanding universe), it is particularly well-suited for teaching about general techniques of ``reading'' and interpreting a spacetime metric 

Consider undergraduate courses where students are introduced to selected concepts and results from general relativity without exposing them to the full mathematical formalism  Such courses have the advantage of introducing students to one of the two great fundamental theories of 20th century physics early on (the other being quantum mechanics); they also profit from subject matter that meets with considerable interest from students \cite{Hartle2006} Using the terminology of Christensen and Moore,\cite{Christensen2012} in the ``calculus only'' approach pioneered by Taylor and Wheeler,\cite{Taylor2001,Taylor2018} spacetime metrics are not derived, but taken as given, and the focus is on learning how to interpret a given spacetime metric  Similar presentations can be found in the first part of the ``physics first'' approach exemplified by Hartle's text book,\cite{Hartle2003} where the concepts of the metric and of geodesics are introduced early on, and their physical consequences explored, while the mathematics necessary for the Einstein equations is only introduced at a later stage  

Whenever the approach involves an exploration of simple metrics such as the Schwarzschild solution, but stops short of the formalism required for the full tensorial form of Einstein's equations, access to a simple derivation of the Schwarzschild solution that does not make use of the advanced formalism can be a considerable advantage 

Simplified derivations of the Schwarzschild solution have a long tradition within general relativity education,\cite{Schiff1960,Harwit1973} although specific simplifications have met with criticism \cite{Rindler1968} This article presents a derivation which requires no deeper knowledge of the formalism of differential geometry beyond an understanding of how to interpret a given spacetime metric $\mathrm{d} s^2$  The derivation avoids the criticism levelled at attempts to derive the Schwarzschild solution from the Einstein equivalence principle in combination with a Newtonian limit,\cite{Gruber1988} relying as it does on a simplified version of the vacuum Einstein equation 

More specifically, I combine the restrictions imposed by the symmetry with the simple form of Einstein's equations formulated by Baez and Bunn \cite{BaezBunn2005} That same strategy was followed by Kassner in 2017,\cite{Kassner2017} but in this text, I use the ``infalling coordinates'' that are commonly associated with the Gullstrand-Painlev\'e form of the Schwarzschild metric,\cite{Martel2001,Visser2005,HamiltonLisle2008} not the more common Schwarzschild coordinates  That choice simplifies the argument even further  In the end, what is required is no more than the solution of an ordinary differential equation for a single function, which yields to standard methods, to obtain the desired result 

\section{Coordinates adapted to spherical symmetry and staticity}
\label{SymmetriesCoordinates}

Assume that the spacetime we are interested in is spherically symmetric and static  In general relativity, a symmetry amounts to the possibility of being able to choose coordinates that are adapted to the symmetry, at least within a restricted sub-region of the spacetime in question  That the spacetime is static is taken to mean that we can introduce a (non-unique) time coordinate ${t}$ so that our description of spacetime geometry does not depend explicitly on ${t}$, and that space and time are completely separate --- in the coordinates adapted to the symmetry, there are no ``mixed terms'' involving $\mathrm{d} {t}$ times the differential of a space coordinate in the metric  If we use ${t}$ to slice our spacetime into three-dimensional hyperplanes, each corresponding to ``space at time ${t}$,'' then each of those 3-spaces has the same spatial geometry  A mixed term would indicate that those slices of space would need to be shifted relative to another in order to identify corresponding points  The mixed term's absence indicates that in adapted coordinates, there is no need for such an extra shift  In those coordinates, we can talk about the 3-spaces as just ``space,'' without the need for specifying which of the slices we are referring to 

In the case of spherical symmetry, we can introduce spherical coordinates that are adapted to the symmetry: a radial coordinate $r$ and the usual angular coordinates $\vartheta,\varphi$, so that the spherical shell at constant $r$ has the total area $4\pi r^2$  In consequence, the part of our metric involving $\mathrm{d}\vartheta$ and $\mathrm{d}\varphi$ will have the standard form
\begin{equation}
r^2(\mathrm{d}\vartheta^2+\sin^2\theta\mathrm{d}\varphi^2) \equiv r^2\mathrm{d}\Omega^2,
\end{equation}
where the right-hand side defines $\mathrm{d}\Omega^2$, the infinitesimal solid angle corresponding to each particular combination of $\mathrm{d}\vartheta$ and $\mathrm{d}\varphi$ 

The radial coordinate slices space into spherical shells, each corresponding to a particular value $r=const $ The rotations around the origin, which are the symmetry transformations of spherical symmetry, map each of those spherical shells onto itself, and they leave all physical quantities that do not explicitly depend on $\vartheta$ or $\varphi$ invariant 

In what follows, we will use the basic structures introduced in this way --- the slices of simultaneous ${t}$, the radial directions within each slice, the angular coordinates spanning the symmetry--adapted spherical shells of area $4\pi r^2$ --- as auxiliary structures for introducing spacetime coordinates  For now, let us write down the shape that our metric has by simple virtue of the spherical symmetry, the requirement that the spacetime be static, and the adapted coordinates, namely
\begin{equation}
\mathrm{d} s^2 = -c^2F(r) \mathrm{d} {t}^2 + G(r) \mathrm{d} r^2 + r^2\:\mathrm{d}\Omega^2  
\label{StaticForm}
\end{equation}
Students familiar with  ``reading'' a spacetime metric will immediately recognize the sign difference between the parts describing space and describing time that is characteristic for spacetime, and the speed of light $c$ that gives us the correct physical dimensions  That there is no explicit dependence on $\varphi$ and $\vartheta$ in the remaining functions $F$ and $G$ is a direct consequence of spherical symmetry  That the factor in front of $\mathrm{d}\Omega^2$ is $r^2$ is a consequence of our coordinate choice, with spherical angular coordinates so that the area of a spherical surface of constant radius $r$ is $4\pi r^2$  That there is no explicit dependence on ${t}$ is one consequence of the spacetime being static; the absence of the mixed term $\mathrm{d} {t}\cdot \mathrm{d} r$ is another  We are left with two unknown functions $F(r)$ and $G(r)$  In the following, let us call ${t}$ and $r$ the {\em static coordinates}  
 
Note that, since $G(r)$ is as yet undefined, we have not yet chosen a specific physical meaning for the length measurements associated with our $r$ coordinate  But because of the $\mathrm{d}\Omega^2$ part, it is clear that whatever choice we make, the locally orthogonal lengths $r\cdot\mathrm{d}\vartheta$ and $r\cdot\sin\vartheta\cdot\mathrm{d}\varphi$ will have the same physical interpretation as for the length measurement corresponding to $\mathrm{d} r$ 

\section{Infalling observer coordinates}
\label{Sec:InfallingObservers}

Now that we know what the radial directions are, at each moment of time ${t}$, we follow Visser\cite{Visser2005} as well as Hamilton and Lisle\cite{HamiltonLisle2008} in defining a family of radially infalling observers   Observers in that family are in free fall along the radial direction, starting out at rest at infinity: In mapping each observer's radial progression in terms of the static coordinate time ${t}$, we adjust initial conditions, specifically: the choice of initial speed at some fixed time ${t}$, in just the right way that the radial coordinate speed goes to zero for each observer in the same way as $r\to\infty $

It is true that talking about ``infalling'' observers already reflects our expectation that our solution should describe the spacetime of a spherically symmetric mass   As we know from the Newtonian limit, such a mass attracts test particles in its vicinity  It should be noted, though, that all our calculations would also be compatible with the limit of no mass being present  In that case, ``infalling'' would be a misnomer, as our family of observers would merely hover in empty space at unchanging positions in $r$  

We can imagine infinitesimal local coordinate systems associated with our observers --- think of the observer mapping out space and time by defining three orthogonal axes, and by measuring time with a co-moving clock  We assume all such little coordinate systems to be non-rotating --- otherwise, we would break spherical symmetry, since rotation would locally pick out a plane of rotation that is distinguishable from the other planes  The radial direction is a natural choice for the first space axis of those little free-falling systems  The other directions, we take to point to observers falling side by side with our coordinate-defining observer --- and to remain pointed at a specific such other observer, once the choice of direction is made 

We assume our infalling observers' clocks to be synchronised at some fixed radius value $r$  By spherical symmetry, those clocks should then be synchronised at {\em all} values of $r$  Anything else would indicate direction-dependent differences for the infalling observers and their clocks, after all  Hence, at any given static time ${t}$, all the infalling observers who are at radius value $r$ show the same proper time $T$ on the ideal clocks travelling along with them  

Once our definition is complete, our static, spherically symmetric spacetime is filled with infalling observers from that family: Whenever we consider an event $\cal E$, there will be an observer from that family passing by at that time, at that location  

Now, consider the coordinate speed of those infalling observers  If we position ourselves at some constant radius value $r$ and watch the falling observers fly by, then we can express both their proper time rate and their coordinate speed in the $r$ direction in terms of $r$ and ${t}$  We can combine the two pieces of information to obtain the rate of change in radial position $r$ with proper time $T$ for those infalling observers  But since the initial conditions for those observers are the same, and since our spacetime is, by assumption, static, the resulting function can only depend on $r$, and not explicitly on ${t}$  Let us rescale that function with the speed of light to make it dimensionless, give it an overall minus sign to make it positive for infalling particles, and call it $\beta(r)$,
\begin{equation}
\beta(r)\equiv -\frac{1}{c}\frac{\mathrm{d} r}{\mathrm{d} T}(r) 
\label{betaDefinition}
\end{equation}

Recall from section \ref{SymmetriesCoordinates} that we also still have the freedom to decide on the physical meaning of $r$  We make the choice of making $\mathrm{d} r$ the physical length measured by one of our infalling observers at the relevant location in spacetime, at constant time $T$  Via our angular coordinates, that implies that length measurements orthogonal to the radial direction, $r\cdot\mathrm{d}\vartheta$ and $r\cdot\sin\vartheta\:\mathrm{d}\varphi$ inherit the same physical interpretation 

As a next step, we transform our metric (\ref{StaticForm}) from the static form into the form appropriate for our coordinate choice $r$ and $T$  We do so by writing the static time coordinate as a function ${t}(T,r)$ in terms of infalling observer time and radius value  In consequence,
\begin{equation}
\mathrm{d} {t} = \frac{\partial{t}}{\partial T}\cdot\mathrm{d} T+ \frac{\partial {t}}{\partial r}\cdot\mathrm{d} r,
\end{equation}
and our new metric now has the form
\begin{align}
 \mathrm{d} s^2 = {} & -c^2 F(r)\left(\frac{\partial t}{\partial T}\right)^2\mathrm{d} T^2 \nonumber \\[0 2em]
 & -2c^2F(r)\left(\frac{\partial t}{\partial T}\right)\left(\frac{\partial t}{\partial r}\right)\mathrm{d} T\:\mathrm{d} r \nonumber \\[0 2em]
 & +\left[G(r)-c^2F(r)\left(\frac{\partial t}{\partial r}\right)^2\right]\mathrm{d} r^2+r^2\:\mathrm{d}\Omega^2 
 \end{align}
At face value, this looks like we are moving the wrong way, away from simplification, since we now have more functions, and they depend on two variables instead of one 

But in fact, this new formulation paves the way for an even simpler form of the metric  Consider a specific event, which happens at given radius value $r$  In a small region around that event, we will introduce a new coordinate $\bar{r}$ to parametrize the radial direction  We want this coordinate to be co-moving with our infalling observers at $r$; each such observer then has a position $\bar{r}=const $ that does not change over time  

Key to our next step is that we {\em know} the metric for the local length and time measurements made by any one of our free-falling observers  By Einstein's equivalence principle, the metric is that of special relativity  Locally, namely whenever tidal effects can be neglected, spacetime geometry for any non-rotating observer in free fall is indistinguishable from Minkowski spacetime as described by a local inertial system 

Since we have chosen both the time coordinate $T$ and the physical meaning of the radial coordinate $r$ so as to conform with the measurements of the local infalling observer, the transformation between $\bar{r}$ and $r$ is particularly simple: It has the form of a Galilei transformation
\begin{equation}
\mathrm{d}\bar{r}= \mathrm{d} r + \beta(r)c\:\mathrm{d} T 
\label{barRshift}
\end{equation}
In that way, as it should be by definition, radial coordinate differences at constant $T$ are the same in both systems, while for an observer at constant $\bar{r},$ with $\mathrm{d} \bar{r}=0$, the relation between $\mathrm{d} r$ and $\mathrm{d} T$ is consistent with the definition of the function $\beta(r)$ in (\ref{betaDefinition}) 

Are you surprised that this is not a Lorentz transformation, as one might expect from special relativity? Don't be  We are not transforming from one local inertial coordinate system to another  The $T$ is already the time coordinate of the infalling observers, so both coordinate systems have the same definition of simultaneity, and time dilation plays no role in this particular transformation  Also, we have chosen $r$ intervals to correspond to length measurements of the infalling observers, so there is no Lorentz contraction, either  It is the consequence of these special choices that gives the relation (\ref{barRshift}) its simple form 

Last but not least, when we analyse specifically an infinitesimal neighbourhood of the point $r,\vartheta,\varphi$, let us make the choice that directly at our point of interest, we make $\bar{r}$ coincide with $r$  Since before, we had only fixed the differential $\mathrm{d} \bar{r}$, we do have the remaining freedom of choosing a constant offset for $\bar{r}$ that yields the desired result 

By Einstein's equivalence principle, the metric in terms of the locally co-moving coordinates $T,\bar{r},\vartheta,\varphi$ is the spherical-coordinate version of the Minkowski metric,
\begin{equation}
\mathrm{d} s^2 = -c^2\mathrm{d} T^2 + \mathrm{d}\bar{r}^2 + \bar{r}^2\mathrm{d}\Omega 
\end{equation}
This version can, of course, be obtained by taking the more familiar Cartesian-coordinate version
\begin{equation}
\mathrm{d} s^2=-c^2\mathrm{d} T^2 + \mathrm{d} X^2 + \mathrm{d} Y^2 + \mathrm{d} Z^2,
\label{CartesianMinkowski}
\end{equation}
applying the definition of Cartesian coordinates $X,Y,Z$ in terms of spherical coordinates $\bar{r},\vartheta,\varphi$
\begin{equation}
x= \bar{r}\:\sin\vartheta\:\cos\varphi, \;\;
y= \bar{r}\:\sin\vartheta\:\sin\varphi, \;\;
z= \bar{r}\:\cos\vartheta,
\end{equation}
to express $\mathrm{d} X, \mathrm{d} Y, \mathrm{d} Z$ in terms of $\mathrm{d} \bar{r}, \mathrm{d}\vartheta, \mathrm{d}\varphi$, and substitute the result into (\ref{CartesianMinkowski}) 

By noting that we have chosen $\bar{r}$ so that, at the specific spacetime event where we are evaluating the metric, $\bar{r}=r$, while, for small radial coordinate shifts around that location, we have the relation (\ref{barRshift}), we can now write down the same metric in the coordinates $T, r, \vartheta,\varphi$, namely as
\begin{equation}
\mathrm{d} s^2 = -c^2\left[
1-\beta(r)^2
\right] \mathrm{d} T^2+2c\beta(r)\mathrm{d} r\:\mathrm{d} T
+\mathrm{d} r^2+r^2\mathrm{d}\Omega^2 
\label{preMetric}
\end{equation}
Since we can repeat that local procedure at any event in our spacetime, this result is our general form of the metric, for all values of $r$  This, then is the promised simplification: By exploiting the symmetries of our solutions as well as the properties of infalling observers, we have reduced our metric to a simple form with no more than one unknown function of one variable, namely $\beta(r)$ 

So far, what I have presented is no more than a long-form version of the initial steps of the derivation given by Visser in his heuristic derivation of the Schwarzschild metric \cite{Visser2005} In the next section, we will deviate from Visser's derivation 

\section{$\beta(r)$ from tidal deformations}
\label{TidalSection}

In the previous section, we had exploited symmetries and Einstein's equivalence principle  In order to determine $\beta(r)$, we need to bring in additional information, namely the Einstein equations, which link the matter content with the geometry of spacetime  For our solution, we only aim to describe the spacetime metric outside whatever spherically-symmetric matter distribution resides in (or around) the center of our spherical symmetry  That amounts to applying the {\em vacuum Einstein equations} 

More specifically, we use a particularly simple and intuitive form of the vacuum Einstein equations, which can be found in a seminal article by Baez and Bunn:\cite{BaezBunn2005} Consider a locally flat free-fall system around a specific event $\cal E$, with a time coordinate $\tau$, local proper time, where the event we are studying corresponds to $\tau=0$  In that system, describe a small sphere of freely floating test particles, which we shall call a {\em test ball}  The particles need to be at rest relative to each other at $\tau=0$  Let the volume of the test ball be $V(\tau)$  Then the vacuum version of Einstein's equations states that
\begin{equation}
\left \frac{\mathrm{d}^2 V}{\mathrm{d}\tau^2}\right|_{\tau=0} = 0 
\label{EinsteinVacuum}
\end{equation}
In words: If there is no matter or energy inside, the volume of such a test ball remains constant in the first order (those were our initial conditions) and the second order (by eq ~[\ref{EinsteinVacuum}])  

If you are familiar with Wheeler's brief summary of Einstein's equations, ``spacetime grips mass, telling it how to move'' and ``mass grips spacetime, telling it how to curve'',\cite{Wheeler1990} you will immediately recognise that this is a specific way for the structure of spacetime telling the test ball particles how to move  The calculation later in this section provides the second part: It will amount to using (\ref{EinsteinVacuum}) to determine the structure of spacetime, namely the still missing function $\beta(r)$, and that is the way for mass, in this case: for the absence of mass, to tell spacetime how to curve 

Note that equation (\ref{EinsteinVacuum}) also holds true in Newtonian gravity  So in a way, this version of Einstein's equation can be seen as a second-order extension of the usual Einstein equivalence principle: Ordinarily, the equivalence principle is a statement about physics in the absence of tidal forces  Equation (\ref{EinsteinVacuum}) adds to this that the lowest-order correction for tidal forces in a freely falling reference frame is that specified by Newtonian gravity  This makes sense, since by going into a free-fall frame, and restricting our attention to a small spacetime region, we have automatically created a weak-gravity situation  In such a situation, tidal corrections are approximately the same as those described by Newton  This argument can serve as a heuristic justification of (\ref{EinsteinVacuum}) 

In 2017, Kassner made use of the Baez-Bunn form of Einstein's vacuum equation to derive the Schwarzschild solution, starting from what we have encountered as the static form of the metric (\ref{StaticForm}) \cite{Kassner2017} We follow the same general recipe, but using the infalling coordinates introduced in section \ref{Sec:InfallingObservers}, which makes our derivation even simpler 

Consider five test particles in a small region of space  Let the motion of each be the same as for the local representative from our coordinate-defining family of infalling observers  We take the central particle $C$ to be at radial coordinate value $r=R$ at the time of the snapshot shown in Fig ~\ref{TestParticlesOutside}  The other four are offset relative to the central particle: As described in the local inertial system that is co-moving with the central particle, one of the particles is shifted by $\Delta l$ upwards in the radial direction, another downward, while two of the particles are offset orthogonally by the same distance 
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0 5\linewidth]{01-free-fall-particles pdf}
\caption{Five test particles in our spherically-symmetric spacetime}
\label{TestParticlesOutside}
\end{center}
\end{figure}
The $\Delta l$ is meant to be infinitesimally small, so while Fig ~\ref{TestParticlesOutside} is of course showing a rather large $\Delta l$ so as to display the geometry of the situation more clearly, we will in the following only keep terms linear in $\Delta l$  

Consider a generic particle, which moves as if it were part of our coordinate-defining family of infalling observers, and which at the time $T_0$ is at $r=r_0$  By a Taylor expansion, that particle's subsequent movement is given by
\begin{equation}
r(T) = r_0 + \frac{\mathrm{d} r}{\mathrm{d} T}(T_0) \cdot \Delta T +\frac12 \frac{\mathrm{d}^2 r}{\mathrm{d} T^2}(T_0) \cdot \Delta T^2
\label{TaylorREvo}
\end{equation}
where $\Delta T\equiv T-T_0$  We know from (\ref{betaDefinition}) that the derivative in the linear term can be expressed in terms of $\beta(r)$; by the same token,
\begin{equation}
\frac{\mathrm{d}^2 r}{\mathrm{d} T^2} = -c\frac{\mathrm{d}\beta}{\mathrm{d} T}=-c\beta' \frac{\mathrm{d} r}{\mathrm{d} T} = c^2\beta\cdot\beta',
\end{equation}
where the prime denotes differentiation of $\beta$ with respect to its argument  Since, in the following, the product of $\beta$ and its first derivative will occur quite often, let us introduce the abbreviation
\begin{equation}
B(r) \equiv \beta(r)\cdot\beta'(r) 
\label{BigBDefinition}
\end{equation}
With these results, can rewrite the Taylor expansion (\ref{TaylorREvo}) as 
\begin{equation}
r(T) = r_0 -c\beta(r_0)\cdot\Delta T + \frac12 c^2B(r_0)\cdot\Delta T^2 
\label{RadialOrbitTime}
\end{equation}
In order to find $r_C(T)$ for our central particle, we simply insert $r_0=R$ into that expression  If, on the other hand, we want to write down the time evolution for particles $U$ and $D$, let us denote it by $r_{U,D}(T)$, we need to evaluate the expression (\ref{RadialOrbitTime}) at the initial location $r_0=R\pm\Delta l$  Since $\Delta l$ is small, we can make a Taylor expansion of $\beta(r)$ and its derivative around $r=R$, and neglect everything beyond the terms linear in $\Delta l$  The result is
\begin{multline}
r_{U,D}(T)=R \pm\Delta l-c\left[
\beta(R)\pm\beta'(R)\Delta l
\right]\Delta T \\[0 2em]
+\frac{c^2}{2}\big[
B(R)\pm B'(R)\Delta l
\big]\Delta T^2
\end{multline}
In consequence, the distance between the upper and lower particle, $d_{\parallel}(T)\equiv r_U(T)-r_D(T),$ changes over time as
\begin{equation}
d_{\parallel}(T) =  2\Delta l\left[
1-c\beta'(R)\Delta T+\frac12c^2 B'(R)\Delta T^2
\right] 
\label{dParallel}
\end{equation}
Next, let us look at how the distance between the particles $L$ and $R$ changes over time  The initial radial coordinate value for each of the particles is
\begin{equation}
r(T_0) = \sqrt{R^2+\Delta l^2}=R\left[1+\frac12\left(\frac{\Delta l}{R}\right)^2\right]\approx R,
\end{equation}
that is, equal to $R,$ as long as we neglect any terms that are higher than linear in $\Delta l$  In consequence, $r_{L,R}(t)$ is the same function as for our central particle, given by eq ~(\ref{RadialOrbitTime}) with $r_0=R$  The transversal (in Fig ~\ref{TestParticlesOutside}: horizontal) distance $d_{\perp}(T)$ between the particles $L$ and $R$ changes in proportion to the radius value,
\begin{align}
d_{\perp}(T) &= 2\Delta l\cdot\frac{r_{L}(T)}{R} \nonumber \\
                 &=2\Delta \left[1-\frac{c\beta(R)}{R}\Delta T+\frac{c^2}{2}\frac{B(R)}{R}\Delta T^2\right] 
                \label{dPerp}
\end{align}
With these preparations, consider the vacuum Einstein equation (\ref{EinsteinVacuum}) for the volume of a test ball  Initially, our particles $C, U, D, L, R$ define a circle, which is deformed to an ellipse  By demanding rotational symmetry around the radial direction, we can construct the associated ellipsoid, which is initially a spherical surface  That ellipsoid has one axis in radial direction, whose length is $d_{\parallel}(T)$, and two axes that are transversal and each have the length  $d_{\perp}(t)$  But that ellipsoid is not quite yet the test ball we need  After all, the particles of the test ball need to be at rest initially, at time $T_0$, in the co-moving system defined by the central particle $C$  Our defining particles are not, as the terms linear in $\Delta T$ in both (\ref{dParallel}) and (\ref{dPerp}) show, where the coefficients of $\Delta T$ correspond to the particles' initial velocities  

In order to define our test ball, we need to consider particles at the same location, undergoing the same acceleration, but which are initially at rest relative to the central particle $C$  

We could go back to the drawing board, back to Fig ~\ref{TestParticlesOutside}, make a more general Ansatz that includes initial velocities which measure the divergence of the motion of our test ball particles from that of the infalling-observer particles, and repeat our calculation while including those additional velocity terms  But there is a short-cut  The only consequence of those additional velocity terms will be to change the terms linear in $\Delta T$ in equations (\ref{dParallel}) and (\ref{dPerp})  And we already know the end result: We will choose the additional terms so as to cancel the terms linear in $\Delta T$ in the current versions of (\ref{dParallel}) and (\ref{dPerp})  But by that reasoning, we can skip the explicit steps in between, and write down the final result right away  The time evolution of the radial-direction diameter of our test ball, let us call it $L_{\parallel}(T)$, must be the same as $d_{\parallel}(T)$, but without the term linear in $\Delta T$  Likewise, the time evolution $L_{\perp}(T)$ of the two transversal diameters must be equal to $d_{\perp}(T)$, but again without the term linear in $\Delta T$  The result is
\begin{align}
L_{\parallel}(T)  &=  2\Delta l \left[1+\frac12c^2B'(R)\Delta T^2\right] \\
L_{\perp}(T) &= 2\Delta l \left[1+\frac{c^2}{2}\frac{B(R)}{R}\Delta T^2\right] 
\end{align}
Thus, our test ball volume is
\begin{align}
V(T) &= \frac{\pi}{6}L_{\parallel}(T) L_{\perp}^2(T) \\
       &= \left \frac{4\pi}{3}\Delta l^3\left[1+{c^2}\left( \frac{B(r)}{r} + \frac{B'(r)}{2}\right)\Delta T^2\right]\right|_{r=R}
\end{align}
For the second time derivative of $V(T)$ to vanish at the time $T=T_0$, we must have
\begin{equation}
\frac{B(r)}{r} + \frac{B'(r)}{2}= 0
\label{VolumeConditionR}
\end{equation}
for all values of $r$  This is readily solved by the standard method of separation of variables: We can rewrite (\ref{VolumeConditionR}) as
\begin{equation}
\frac{\mathrm{d} B}{B} = -2\frac{\mathrm{d} r}{r},
\end{equation}
which is readily integrated to give
\begin{equation}
\ln(B) = -\ln(r^{2}) + const   \;\; \Rightarrow \;\; \ln(Br^2) = C',
\end{equation}
with a constant $C'$, which upon taking the exponential gives us
\begin{equation}
Br^2= C,
\label{BSolution}
\end{equation}
with a constant $C$  Note that the constant $C$ can be negative --- there is no reason the constant $C'$ needs to be real; only our eventual function $B(r)$ needs to be that, and it is clear that (\ref{BSolution}) satisfies the differential equation
(\ref{VolumeConditionR}) for any constant $C$, positive, zero, or negative  By (\ref{BigBDefinition}), the solution (\ref{BSolution}) corresponds to the differential equation
\begin{equation}
\beta(r)\beta'(r) = \frac{C}{r^2}
\end{equation}
for our function $\beta$; with another separation of variables, we can re-write this as 
\begin{equation}
\beta\cdot\mathrm{d}\beta=C\frac{\mathrm{d} r}{r^2} 
\end{equation}
Both sides are readily integrated up; we can solve the result for $\beta(r)$ and obtain
\begin{equation}
\beta(r) = \sqrt{
-\frac{2C}{r} +2D
},
\end{equation}
where $D$ is the second integration constant, and where we have chosen the proper sign, since we know that $\beta(r)>0$  That brings us to the last step: The requirement that, for large values of $r$, the description provided by our solution should correspond to the results from Newtonian gravity  First of all, we note that our initial condition for the infalling observers, which had those observers start out at zero speed at infinity, means that we must choose $D=0$  Then, as we would expect, $\beta(r)$ for large values of $r$ becomes very small, corresponding to small speeds  But at slow speeds, time and length intervals as measured by the infalling observer will become arbitrarily close to time and length intervals as measured by an observer at rest in our static coordinate system at constant $r$, using the static time coordinate ${t}$  As is usual, we identify these coordinates with those of an approximately Newtonian description  In that description, the radial velocity is
\begin{equation}
v(r) = \sqrt{\frac{2GM}{r}},
\end{equation}
which follows directly from energy conservation for the sum of each observer's kinetic and Newtonian-gravitational potential energy  This fixes the remaining integration constant as
\begin{equation}
C = -\frac{GM}{c^2},
\end{equation}
and the final form of our function $\beta(r)$ becomes
\begin{equation}
\beta(r) = \sqrt{\frac{2GM}{rc^2}} 
\end{equation}
Inserting this result in (\ref{preMetric}), we obtain the metric
\begin{equation}
\mathrm{d} s^2 = -c^2\left[
1-\frac{2GM}{rc^2}
\right]\mathrm{d} T^2+2\sqrt{\frac{2GM}{r}}\mathrm{d} r\:\mathrm{d} T+\mathrm{d} r^2+r^2\mathrm{d}\Omega^2 
\label{GPMetric}
\end{equation}
This is known as the Gullstrand-Painlev\'e version of the Schwarzschild metric \cite{Martel2001,Visser2005,HamiltonLisle2008} A  last transformation step brings us back to the traditional Schwarzschild form  Recall our discussion in sec ~\ref{SymmetriesCoordinates}, leading up to the explicitly static form (\ref{StaticForm}) of the metric? The main difference between our current form and the static version is the mixed term containing $\mathrm{d} r\:\mathrm{d} T$ in (\ref{GPMetric})  Everything else already has the required shape  Inserting the Ansatz
\begin{equation}
\mathrm{d} T = \mathrm{d} t + \xi(r) \mathrm{d} r
\end{equation}
into the metric (\ref{GPMetric}), it is straightforward to see that the mixed term vanishes iff our transformation is
\begin{equation}
\mathrm{d} T = \mathrm{d} t +\frac{\sqrt{2GM/r}}{c^2\left(1-\frac{2GM}{rc^2}\right)}\mathrm{d} r 
\label{TtTrafo}
\end{equation}
Substitute this into (\ref{GPMetric}), and the result is the familiar form of the Schwarzschild metric in Schwarzschild's original coordinates $t,r,\vartheta,\varphi$, 
\begin{equation}
\mathrm{d} s^2 = -c^2\left(1-\frac{2GM}{c^2 r}
\right)\mathrm{d} t^2 + \frac{\mathrm{d} r^2}{\left(1-\frac{2GM}{c^2 r}
\right)} + r^2\mathrm{d}\Omega^2 
\end{equation}

\section{Conclusion}
Using coordinates adapted to the symmetries, we were able to write down the spherically symmetric, static spacetime metric  On this basis, and using the family of infalling observers that is characteristic for the Gullstrand-Painlev\'e solution, we wrote down the metric in the form (\ref{preMetric}), with a single unknown function $\beta(r)$  From the simplified form (\ref{EinsteinVacuum}) of the vacuum Einstein equations, as applied to a test ball in free fall alongside one of our family of observers, we were able to determine $\beta(r)$, up to two integration constants  By using the Einstein equation, we escape the restrictions imposed on simplified derivations by Gruber et al \cite{Gruber1988} 

From the initial condition for our infalling observers, as well as from the Newtonian limit at large distances from our center of symmetry, we were able to fix the values of the two intergration constants  Our derivation does not require knowledge of advanced mathematical concepts beyond the ability to properly interpret a given metric line element $\mathrm{d} s^2$  Even our analysis of tidal effects proceeds via a simple second-order Taylor expansion, leading to differential equations for $\beta(r)$ that are readily solved using two applications of the method of separation of variables  

What is new about the derivation presented here is the combination of the Baez-Bunn equations with the infalling coordinates typical for the Gullstrand-Painlev\'e form of the metric --- this combination is what, in the end, makes our derivation particularly simple  In turn, this simplicity is what should make the derivation particularly useful in the context of teaching general relativity in an undergraduate setting 

The derivation proceeds close to the physics, and gives ample opportunity to discuss interesting properties of Einstein's theory of gravity  Students who are presented with this derivation, either as a demonstration or as a (guided) exercise, will come to understand the way that symmetries determine the form of a metric, the deductions that can be made from Einstein's equivalence principle, and last but not least that we need to go beyond the equivalence principle, and consider tidal forces, to completely define our solution 

\section*{Acknowledgements}

I would like to thank Thomas M\"uller for helpful comments on an earlier version of this text 


Context 2: Current address: Division of Brain Sciences, Department of Medicine, Imperial College London, London, United Kingdom 
In a variety of species, reduced food intake, and in particular protein or amino acid (AA) restriction, extends lifespan and healthspan  However, the underlying epigenetic and/or transcriptional mechanisms are largely unknown, and dissection of specific pathways in cultured cells may contribute to filling this gap  We have previously shown that, in mammalian cells, deprivation of essential AAs (methionine/cysteine or tyrosine) leads to the transcriptional reactivation of integrated silenced transgenes, including plasmid and retroviral vectors and latent HIV-1 provirus, by a process involving epigenetic chromatic remodeling and histone acetylation  Here we show that the deprivation of methionine/cysteine also leads to the transcriptional upregulation of endogenous retroviruses, suggesting that essential AA starvation affects the expression not only of exogenous non-native DNA sequences, but also of endogenous anciently-integrated and silenced parasitic elements of the genome  Moreover, we show that the transgene reactivation response is highly conserved in different mammalian cell types, and it is reproducible with deprivation of most essential AAs  The General Control Non-derepressible 2 (GCN2) kinase and the downstream integrated stress response represent the best candidates mediating this process; however, by pharmacological approaches, RNA interference and genomic editing, we demonstrate that they are not implicated  Instead, the response requires MEK/ERK and/or JNK activity and is reproduced by ribosomal inhibitors, suggesting that it is triggered by a novel nutrient-sensing and signaling pathway, initiated by translational block at the ribosome, and independent of mTOR and GCN2  Overall, these findings point to a general transcriptional response to essential AA deprivation, which affects the expression of non-native genomic sequences, with relevant implications for the epigenetic/transcriptional effects of AA restriction in health and disease 
Copyright: © 2018 De Vito et al  This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited 
Data Availability: All relevant data are within the paper and its Supporting Information files  RNAseq data are available in the ArrayExpress database under the accession number E-MTAB-6452 
Funding: This study was funded by the Ajinomoto Innovation Alliance Program, (AIAP; https://www ajinomoto com/en/rd/AIAP/index html#aiap) (to M V S and D G), which is a joint research initiative of Ajinomoto Co , Inc , Japan  One of the authors [M B ] is an employee of Ajinomoto Co , and his specific roles are articulated in the ‘author contributions’ section  The commercial funder provided support in the form of salary for author [M B ] and some of the necessary research materials (medium for cell culture), but did not have any additional role in the study design, data collection and analysis, or preparation of the manuscript, and the authors had unrestricted access to the data  Due to a confidentiality agreement, the commercial funder participated only in the decision to publish the data obtained during the study, without any restriction 
Competing interests: This study was funded by Ajinomoto Co , Inc , Japan and one of the authors [M B ] is an employee of this commercial funder  No other employment or consultancy relationships exist with the commercial funder, and no patents, products in development, or marketed products result from this study  The authors declare that no competing interests exist and that the commercial affiliation of one of the authors does not alter the adherence of authors to all PLOS ONE policies on sharing data and materials 
In animals, excessive, insufficient, or imbalanced nutrient availability is known to strongly impact on phenotype and health, both short and long-term, and across generations [1, 2]  In particular, studies in yeast, animal models and humans have shown that reduced food intake, reducing either overall calories, or only sugars, proteins, or even single amino acids (AA), such as Methionine (Met), may extend lifespan and healthspan, and reduce the risk of cancer and other age-related diseases [3–9]  In addition, fasting or specific AA deprivation have shown potential therapeutic applications, owing to their ability to directly reduce the growth of some tumor types [10, 11], sensitize cancer cells to chemo- or immunotherapy [12, 13], and allow efficient hematopoietic stem cell engraftment   However, little is known about the specific processes and molecular mechanisms mediating the roles of nutrient restriction in human health and longevity 
A properly balanced diet in metazoans contains optimal amounts of a subset of AA, which cannot be synthetized de novo and are therefore named essential amino acids (EAAs)  In humans these include Met, Histidine (His), Isoleucine (Ile), Leucine (Leu), Lysine (Lys), Phenylalanine (Phe), Threonine (Thr), Tryptophan (Trp), and Valine (Val), while a few others are considered as semi-essential, such as Glutamine (Gln) and Tyrosine (Tyr) [15, 16]  Consistently, EAA deprivation triggers a cell-autonomous adaptive response, characterized by extensive metabolic and gene expression modifications, implementing biosynthetic, catabolic, and plasma membrane transport processes, aimed at reconstituting the full AA complement [17, 18]  The best known and conserved pathways responding to AA deprivation are triggered by mechanistic Target of Rapamycin Complex 1 (mTORC1) and General amino acid Control Non-derepressible 2 (GCN2) protein kinases [15, 19, 20]  Activation of mTORC1 requires in particular the presence of Gln, Arg and Leu, but also Met , which activate the kinase through sensors mainly acting upstream of Rag GTPases at lysosomal membranes   In turn, mTORC1 promotes cell growth, proliferation and anabolism upon activation, and translational attenuation and autophagy upon inhibition [19, 20] 
By contrast, GCN2 is activated by deprivation of any individual EAA, by means of its histidyl-tRNA synthetase-related domain, which binds uncharged tRNAs accumulating during AA limitation [23, 24]  Upon activation, GCN2 phosphorylates and inhibits its only known downstream target, namely the eukaryotic Initiation Factor 2 α (eIF2α), thereby initiating the Integrated Stress Response (ISR)  This leads to attenuation of general translation, and induction of a transcriptional/translational program, aimed at increasing stress resistance and restoring cell homeostasis, by upregulating a specific subset of genes, including Activating Transcription Factor 4 (ATF4) and C/EBP-Homologous Protein (CHOP) [25–27]  Thus, inhibition of mTORC1 and activation of GCN2 by AA restriction cooperate to attenuate general translation at the initiation step, increase catabolism and turnover, and enhance stress resistance to promote adaptation   However, how these processes eventually induce protective mechanisms against the alterations associated with aging, which include pervasive epigenetic and transcriptional changes [28, 29], remains largely unknown 
We previously reported the unexpected observation that prolonged deprivation of either Tyr, or of both Methionine and Cysteine (Met/Cys), triggers the selective and reversible reactivation of exogenous transcriptional units, including plasmids, retroviral vectors and proviruses, integrated into the genome and transcriptionally repressed by defensive mechanisms against non-native DNA sequences [30, 31]  This phenomenon was observed both in HeLa epithelial and ACH-2 lymphocytic human cells, and was independent of the transgene or provirus (Ocular Albinism type 1, OA1; Green Fluorescent Protein, GFP; Lysosomal-Associated Membrane Protein 1, LAMP1; Human Immunodeficiency Virus-1, HIV-1), or of the exogenous promoter driving their transcription, either viral (cytomegalovirus, CMV; Long Terminal Repeat, LTR) or human (Phospho-Glycerate Kinase 1, PGK1; Elongation Factor-1α, EF-1α)   Furthermore, this transgene reactivation response was not reproduced by serum starvation, activation of p38, or pharmacological inhibitors of mTOR (PP242 or rapamycin), sirtuins and DNA methylation  By contrast, it was induced by pan histone deacetylase (HDAC) inhibitors, and by selective inhibitors of class II HDACs   Consistently, we found that the mechanism responsible involves epigenetic modifications at the transgene promoter, including reduced nucleosome occupancy and increased histone acetylation, and is mediated in part by reduced expression of a class II HDAC, namely HDAC4  
These findings indicate that AA deprivation induces a specific epigenetic and transcriptional response, affecting the expression of newly-integrated exogenous transgenes and proviruses, and suggesting that endogenous sequences sharing similar structural and functional features may represent a transcriptional target as well [30, 31]  In particular, transposable elements, such as LTR-retrotransposons (or endogenous retroviruses, ERVs), are genomic “parasites” anciently-integrated into the genome, and silenced by epigenetic mechanisms of mammalian cells against the spreading of mobile elements, eventually becoming "endogenized" during evolution [32, 33]  This raises the question of whether their expression is also sensitive to AA restriction  In addition, it remains unclear whether or not the transgene reactivation response is related to specific AA deprivations, and most importantly which is the AA sensing/signaling pathway involved, in particular whether the GCN2 kinase is implicated  Thus, here we used the reactivation of silenced transgenes in cultured cells, as a model to investigate a novel molecular pathway induced by imbalanced EAA starvation, implicated in the epigenetic/transcriptional regulation of exogenous non-native DNA sequences and possibly of other endogenous anciently-integrated genomic elements 
HeLa human epithelial carcinoma, HepG2 human hepatocellular carcinoma and C2C12 mouse skeletal muscle cells were maintained in DMEM containing glutaMAX (Invitrogen) and supplemented with 10% FBS (Sigma), 100 U/ml penicillin G (Invitrogen), 100 mg/ml streptomycin (Invitrogen), at 37°C in a 5% CO2 humidified atmosphere  Cell lines carrying integrated and partially silenced transgenes were also maintained in 600–1000 μg/ml G418 
The C2C12 cell line was provided by ATCC  HeLa and HepG2 cells were obtained by Drs  F  Blasi and G  Tonon at San Raffaele Scientific Institute, Milan, Italy, respectively, and were authenticated by Short Tandem Repeat (STR) profiling, using the Cell ID System kit (Promega), according to the manufacturer’s instructions  Briefly, STR-based multiplex PCR was carried out in a final volume of 25 μL/reaction, including 5 μL Cell ID Enzyme Mix 5X, 2 5 μL Cell ID Primer Mix 10X and 3 ng of template DNA  The thermal cycling conditions were: 1 cycle at 96°C for 2 min, followed by 32 cycles at 94°C for 30 sec, 62°C for 90 sec, and 72°C for 90 sec, and 1 cycle at 60°C for 45 sec  The following STR loci were amplified: AMEL, CSF1PO, D13S317, D16S539, D21S11, D5S818, D7S820, TH01, TPOX, vWA  Fragment length analysis of STR-PCR products was performed by Eurofins Genomics, using standard procedures of capillary electrophoresis on the Applied Biosystems 3130 XL sequencing machine, and assessment of the STR profile was performed at the online STR matching analysis service provided at http://www dsmz de/fp/cgi-bin/str html 
Stable cell clones, expressing myc-tagged human OA1 (GPR143) or GFP transcripts, were generated using pcDNA3 1/OA1myc-His or pcDNA3 1/EGFP vectors   Briefly, HeLa, HepG2 and C2C12 cells were transfected using FuGENE 6 (Roche) and selected with 800, 1000, and 650 μg/ml of G418 (Sigma), respectively, which was maintained thereafter to avoid loss of plasmid integration  G418-resistant clones were isolated and analyzed for protein expression by epifluorescence and/or immunoblotting 
Full DMEM-based medium, carrying the entire AA complement, and media deprived of Met/Cys (both AAs), Met (only), Cys (only), Alanine (Ala), Thr, Gln, Val, Leu, Tyr, Trp, Lys and His were prepared using the Nutrition free DMEM (cat #09077–05, from Nacalai Tesque, Inc , Kyoto, Japan), by adding Glucose, NaHCO3, and either all 20 AAs (for full medium) or 18–19 AAs only (for deprivations of two-one AAs)  Single AAs, Glucose, and NaHCO3 were from Sigma  Further details and amounts utilized are indicated in S1 Table  All media were supplemented with 10% dialyzed FBS (Invitrogen), 100 U/ml penicillin G (Invitrogen), 100 mg/ml streptomycin (Invitrogen), and G418 as required  HBSS was from Invitrogen  Cells were seeded at 10–30% of confluency; cells to be starved for 48 h were plated 2–3 times more confluent compared to the control  The following day, cells were washed and cultured in the appropriate medium, with or without EAA, for 24–48 h 
L-Histidinol (HisOH), PP242, Integrated Stress Response Inhibitor (ISRIB), SP600125, Cycloheximide (CHX) were from Sigma; Salubrinal was from Tocris Bioscience; U0126 was from Promega  Drugs were used at the following final concentrations: HisOH at 4–16 mM; PP242 at 1–3 μM; ISRIB at 100 nM; SP600125 at 20 μM in HepG2 cells and 50 μM in HeLa cells; Cycloheximide (CHX) at 50 ug/ml in HepG2 cells and 100 ug/ml in HeLa cells; Salubrinal at 75 μM; U0126 at 50 μM  Vehicle was used as mock control  Treatments with drugs to be tested for their ability to inhibit transgene reactivation (ISRIB, SP600125 and U0126) were initiated 1h before the subsequent addition of L-Histidinol (ISRIB) or the subsequent depletion of Met/Cys (SP600125 and U0126) 
Total RNA was purified using the RNeasy Mini kit (Qiagen), according to manufacturer’s instructions  RNA concentration was determined by Nanodrop 8000 Spectrophotometer (Thermo Scientific)  Equal amount (1 μg) of RNA from HeLa, HepG2 and C2C12 cells was reverse transcribed using the SuperScript First-Strand Synthesis System for RT-PCR (Invitrogen) using oligo-dT as primers, and diluted to 5 ng/μl  The cDNA (2 μl) was amplified by real-time PCR using SYBR green Master Mix on a Light Cycler 480 (Roche), according to manufacturer’s instructions  The thermal cycling conditions were: 1 cycle at 95°C for 5 min, followed by 40–45 cycles at 95° for 20 sec, 56° for 20 sec and 72° for 20 sec  The sequences, efficiencies and annealing temperatures of the primers are provided in S2 Table  Data were analyzed with Microsoft Excel using the formula EtargetΔct target (control-sample) /EreferenceΔct reference (control-sample)   Reference genes for normalizations were ARPC2 (actin-related protein 2/3 complex, subunit 2) for HeLa and HepG2 cells; and Actb (actin beta) for C2C12 cells, unless otherwise indicated 
siRNA (Mission esiRNA, 200 ng/μL; Sigma) against ATF4 and GCN2 were designed against the targeted sequences NM_001675 and NM_001013703, respectively  Cells seeded in 6-well plates were transfected with 1 μg of siRNAs and 5 μL of Lipofectamine 2000 (Invitrogen), following manufacturer’s instructions, at day 1 post-plating for ATF4 and at day 1 and 2 post-plating for GCN2  At day 2 (ATF4) or 3 (GCN2) post-plating, cells were washed and cultured in medium in the absence or presence of HisOH 4 mM for 6 h  siRNAs against RLuc (Sigma), targeting Renilla Luciferase, were used as negative control  For CRISPR/Cas9 experiments, we used the “all-in-one Cas9-reporter” vector, expressing GFP (Sigma), which is characterized by a single vector format including the Cas9 protein expression cassette and gRNA (guide RNA)  GFP is co-expressed from the same mRNA as the Cas9 protein, enabling tracking of transfection efficiency and enrichment of transfected cells by fluorescence activated cell sorting (FACS)  The human U6 promoter drives gRNA expression, and the CMV promoter drives Cas9 and GFP expression  The oligonucleotide sequences for the three gRNAs targeting GCN2 exon 1 or 6 are listed in S2 Table  We transfected HeLa and HepG2 cells with these plasmids individually (one plasmid one guide) and sorted the GFP-positive, transfected cells by FACS  Screening GCN2-KO clones was performed by western blotting  In the case of HepG2-OA1 cells, two rounds of selection were necessary to obtain three GCN2-KO clones by using a guide RNA against exon 1  Compared to the original HepG2-OA1 cell line and to the clone resulting from the first round of selection (185#27), the selected clones E23, F22 and F27 showed a very low amount—if any—of residual GCN2 protein (see results) 
Genomic DNA of HeLa and HepG2 cells was purified using DNeasy Blood and Tissue kit (Qiagen), according to the manufacturer’s instructions  DNA concentration was determined by Nanodrop 8000 Spectrophotometer (Thermo Scientific)  PCR conditions for amplification of GCN2 exon 1 and 6 were as follows: 1 cycle at 94°C for 5 min, followed by 35 cycles at 94°C for 40 sec, 56°C for 40 sec, and 72°C for 40 sec; and a final extension step of 5 min at 72°C  The primer sequences are provided in S2 Table 
For OA1, western immunoblotting was carried out as described   For GCN2, cells were lysed in RIPA buffer, boiled at 95°C for 5 min and resolved on a 7 5% polyacrylamide gel; immunoblotting was then performed following standard procedures  Primary Abs were as follows: anti-human OA1, previously developed by our group in rabbits ; anti-GCN2 (Cell Signaling, Cat  #3302) 
Statistical analyses were performed using Microsoft Excel for Mac (version 15 32, Microsoft) for Student’s t-test; or GraphPad Prism (version 5 0d for Mac, GraphPad Software, Inc ) for one-way analysis of variance (ANOVA), followed by Dunnett’s or Tukey’s multiple comparisons post-tests  T-test was used when only two means, typically sample versus control, were compared, as specified in the figure legends  One way ANOVA was used for multiple comparisons, followed by either a Dunnett’s (to compare every mean to a control mean), or a Tukey’s (to compare every mean with every other mean) post-test, by setting the significance level at 0 05 (95% confidence intervals)  Both tests compare the difference between means to the amount of scatter, quantified using information from all the groups  Specifically, Prism computes the Tukey-Kramer test, allowing unequal sample sizes  P values in Figures are generally referred to comparison between a sample and the control (full medium/mock), and are indicated as follows: *P<0 05, **P<0 01, ***P<0 001  Comparisons not involving the control are similarly indicated, by a horizontal line at the top of the graphs, encompassing the two samples under analysis  Additional details regarding the specific experiments are reported in the Figure Legends 
To examine the expression behavior of genomic repeats upon AA starvation, we performed a transcriptomic analysis taking advantage of an intramural sequencing facility  HeLa-OA1 cells were cultured in normal medium (for 6-30-120 hours) or in absence of Met/Cys (for 6-15-30-72-120 hours)  Total RNA was prepared using Trizol (Sigma) to preserve transcripts of both small and long sizes (from Alu, of about 0 3 kb, to Long Interspersed Nuclear Elements, LINEs, and ERVs, up to 6–8 kb long), DNase treated to avoid contamination of genomic DNA, and processed for NGS sequencing by Ovation RNA-Seq System V2 protocol and HiSeq 2000 apparatus  Raw sequence data (10–20 M reads/sample) were aligned to the human genome (build hg19) with SOAPSplice   Read counts over repeated regions, defined by RepeatMasker track from UCSC genome browser , were obtained using bedtools suite   Normalization factors and read dispersion (d) were estimated with edgeR , variation of abundance during time was analyzed using maSigPro package , fitting with a negative binomial distribution (Θ = 1/d, Q = 0 01), with a cutoff on stepwise regression fit r2 = 0 7  Read counts were transformed to RPKM for visualization purposes  The expression of the OA1 transgene and HDAC4, which are progressively up- and down-regulated during starvation, respectively , were used as internal controls 
For genomic repeat analysis, reads belonging to repetitive elements were classified according to RepeatMasker and assigned to repeat classes (total number in the genome = 21), families (total number in the genome = 56) and finally subfamilies (total number in the genome = 1396), each including a variable number of genomic loci (from a few hundred for endogenous retroviruses, up to several thousand for Alu)  Repeat subfamilies were then clustered according to their expression pattern in starved vs control cells, by maSigPro using default parameters, and repeats classes or families that are significantly enriched in each cluster, compared to all genomic repeats, were identified by applying a Fisher Exact test (using scipy stats, a statistical module of Python)  Alternatively, differentially expressed repeat subfamilies were identified by averaging three time points of starvation (15-30-72 h) and controls  Repeats significantly up- or downregulated (104 and 77, respectively) were selected based on a P value <0 05 (unpaired two-tailed Student’s t-test, assuming equal variance), and analyzed for their class enrichment by a Fisher Exact test as described above 
For gene set enrichment analysis of Met/Cys deprived vs control HeLa cells, differentially expressed genes were selected considering three time points of starvation (15-30-72 h) and controls, based on a P value <0 05 (unpaired two-tailed Student’s t-test, assuming equal variance) and a fold change >2  This led to a total of 2033 differentially expressed genes, 996 upregulated and 1037 downregulated  The enrichment analysis was performed separately for up and down regulated genes, or with all differentially expressed genes together (both), using the KEGG database  The analysis was performed with correction for the background of all expressed genes (about 13600 genes showing an average expression over 3 starvation and 3 control samples of at least 5 counts) and by using default parameters (adjusted P value and q-value cut-off of <0 05 and 0 2, respectively)  Differentially expressed genes were also selected considering all starvation time points, as with genomic repeats, by maSigPro using default parameters, and a fold change of at least 1 5, leading to similar enrichment results (not shown)  RNAseq gene expression data are available in the ArrayExpress database under the accession number E-MTAB-6452 
To provide proof-of-principle that AA starvation may affect the expression of transposable elements, we performed an RNAseq analysis of the previously described HeLa-OA1 cells, carrying an integrated and partially silenced OA1 transgene   Since the reactivation of the transgene by starvation is a progressive phenomenon , we performed a time-course experiment, where each time point represents one biological sample, rather than a biological triplicate of a single time point  To this aim, cells were cultured either in normal medium, or in absence of Met/Cys for different time points (6-15-30-72-120 hours), resulting in the progressive upregulation of the OA1 transgene during starvation (Fig 1A and 1B), consistent with previously published results   The expression of genomic repeats was determined according to RepeatMasker annotation and classification into classes, families, and subfamilies  Repeat species were then subjected to differential expression and enrichment analyses in starved vs control conditions  Out of 1396 annotated repeat subfamilies, 172 species displayed a differential expression profile during starvation 
Fig 1  Exogenous transgene and endogenous retroviruses are upregulated in Met/Cys-deprived HeLa cells 
(A,B) Exogenous integrated transgene (OA1) mRNA abundance in HeLa-OA1 cells, cultured in Met/Cys-deprived medium for the indicated time points, and analyzed by RNAseq (A), or RT-qPCR (B), compared to full medium  Data represent RPKM (A), or mean ± SD of 2 technical replicates, expressed as fold change vs  control (full medium at 6 h = 1) (B)  (C) Clustering of 172 genomic repeat subfamilies, differentially expressed upon starvation, according to their expression profile  (D) Class distribution of repeat subfamilies belonging to differential expression clusters, compared to all genomic repeat subfamilies (first column)  Class DNA includes DNA transposons; SINE includes Alu; LINE includes L1 an L2; LTR includes endogenous retroviruses and solitary LTRs; Satellite includes centromeric acrosomal and telomeric satellites; Others includes SVA, simple repeats, snRNA, and tRNAs  LTR-retroelements are significantly enriched among repeats that are upregulated upon starvation, while LINEs are significantly enriched among repeats that are downregulated  *P<0 05, ***P<0 001 (Fisher exact test) 
As shown in Fig 1C, the clustering of differentially expressed repeats, according to their expression pattern, reveals profiles comparable to the behavior of the transgene in the same conditions, i e  upregulation upon starvation and no change in regular medium (Cluster 1 and 2)  In particular, Cluster 1 contains sequences that, similarly to the OA1 transgene, are progressively upregulated upon starvation (Fig 1A and 1C) , while Cluster 2 contains sequences that are upregulated at early time points  Interestingly, repeat families that are significantly enriched in these two clusters belong mostly to the group of LTR-retrotransposons, including ERV1, ERVK, ERVL, ERVL-MaLR and other LTR sequences (Fig 1D; S1A and S2A Figs)  By contrast, DNA transposons (such as TcMar-Tigger) and L1 non-LTR retrotransposons are enriched among repeats that are downregulated during starvation, particularly at late time points (Clusters 3 and 4) (Fig 1D; S1A and S2A Figs)  Consistent results were obtained by selecting significantly up- or downregulated genomic repeats (overall 181 species), based on their average expression out of three time points of starvation (15-30-72 h, when the transgene upregulation is more homogeneous) and controls, and on a P value <0 05 (S1B and S2B Figs)  These findings suggest that EAA starvation induces genome-wide effects involving repetitive elements, and that—among major repeat classes—it upregulates in particular the expression of ERVs 
In addition, to obtain a general overview of main gene pathways changing their expression together with the transgene during AA starvation, we performed gene expression and enrichment analyses of regular genes, by considering three time points of starvation (15-30-72 h) and controls  Differentially expressed genes were selected based on a P value <0 05 and a fold change between means of at least 2, and analyzed with the EnrichR tool   As shown in Fig 2 and S1 File, enrichment analyses against the KEGG and Reactome databases reveals a predominance of downregulated pathways, namely ribosome and translation, proteasome, AA metabolism, oxidative phosphorylation and other pathways related to mitochondrial functions, which are affected in Huntington, Alzheimer and Parkinson diseases (http://www genome jp/kegg/pathway html)  In particular, a large fraction of ribosomal protein mRNAs is downregulated upon Met/Cys starvation (Fig 2A and 2C; S1 File), consistent with the notion that their genes–despite being scattered throughout the genome—are coordinately expressed in a variety of conditions   This reduced expression may depend on multiple pathways that control ribosome biogenesis in response to external stimuli, including the downregulation of Myc activity , the downregulation of mTORC1 [42, 44], or possibly the activation of the ISR, as described in yeast   By contrast, upregulated genes show a significant enrichment for transcription and gene expression (Fig 2B)  Similar results were obtained by the Gene Ontology Biological Process (GO-BP) database (S1 File), overall indicating a general downregulation of translation and metabolism, and upregulation of transcription, during the time interval of Met/Cys starvation corresponding to the transgene upregulation 
Fig 2  Gene set enrichment analysis of Met/Cys-deprived HeLa cells 
Differentially expressed genes between three time points of starvation (15-30-72 h) and controls were selected based on a P value <0 05 and a fold change of at least 2, leading to a total of 996 upregulated, and 1037 downregulated genes  The enrichment analysis was performed separately for up and down regulated genes, using the EnrichR tool and the KEGG (A) and REACTOME (B, C) databases  Ranking is based on the combined score provided by EnrichR, and categories are displayed up to 20 items with an Adjusted P value <0 05  No significant categories were found with upregulated genes against the KEGG database  All data are shown in S1 File  The enrichment analysis using all differentially expressed genes together did not reveal any additional enriched process 
To characterize the pathway leading to the reactivation of silenced transgenes, we used HeLa-OA1 and HeLa-GFP cells, as described   In addition, to test cell types relevant for AA metabolism, such as liver and muscle, we generated clones of HepG2 human hepatoma and C2C12 mouse skeletal muscle cells, stably transfected with plasmids for OA1 and GFP transgenes, respectively (HepG2-OA1 and C2C12-GFP cells; endogenous OA1 is not expressed in any of these cell types)  In all cases, the integrated transgenes are under the control of the CMV promoter in the context of a pcDNA3 1 plasmid, are partially silenced, and can be efficiently upregulated by HDAC inhibitors (trichostatin A, TSA; ref   and S3A, S3B and S4A Figs), indicating that their expression is controlled at least in part by epigenetic mechanisms, as previously described  
To establish whether the reactivation response results from the shortage of specific AAs only, such as Met/Cys, or it is triggered by any AA deprivations, we cultured HeLa-OA1, HeLa-GFP, HepG2-OA1 and C2C12-GFP cells for 24–48 hours with a battery of media deprived of EAAs or semi-EAAs, including Met/Cys, Thr, Gln, Val, Leu, Tyr, Trp, Lys, and His  As negative controls, cells were cultured in full medium, carrying the entire AA complement, and in a medium deprived of Ala, a non-essential AA  The expression of the transgene transcript was then evaluated by RT-qPCR  As shown in Fig 3, and in S3C and S4B Figs, most EAA-deficiencies induced reactivation of the OA1 or GFP transgenes in all four cell lines, with the notable exception of Trp deprivation, which consistently resulted in no or minimal reactivation of the transgenes  Indeed, despite some variability, Met/Cys deficiency, but also Thr, Val, Tyr, and His deprivation always gave an efficient response, while Leu, Gln and Lys elicited evident responses in some cases, but not in others  Depletion of Phe gave results comparable to Tyr deprivation, however it significantly altered multiple reference genes used for normalization and therefore was eventually omitted from the analysis (not shown)  Finally, in the above experiments we used a combined Met/Cys deficiency, to avoid the potential sparing of Met by Cys  and for consistency with our previous studies   Nevertheless, the analysis of single Met or Cys starvation, both at the protein and transcript levels, revealed an exclusive role of Met deprivation in transgene reactivation, consistent with the notion that Cys is not an EAA (S3D and S3E Fig) 
Fig 3  EAA deprivation induces reactivation of silent transgenes in HeLa and HepG2 cells 
Relative transgene (OA1) and CHOP mRNA abundance in HeLa-OA1 (A) and HepG2-OA1 (B) cells, cultured in various AA-deprived media for 48 h and 24 h, respectively, compared to full medium  Mean ± SEM of 3 independent experiments  Data are expressed as fold change vs  control (full medium = 1)  *P<0 05, **P<0 01, ***P<0 001 (one way ANOVA, followed by Dunnett’s post-test vs  full medium) 
Collectively, these results indicate that transgene reactivation by EAA starvation is reproducible with most EAAs, shared by different cell types (epithelium, liver, and skeletal muscle), and conserved in different mammalian species (human, mouse) 
mTORC1 inhibition and GCN2 activation trigger the best-known signaling pathways responding to AA starvation   We previously showed that inhibition of mTORC1 is not sufficient to reproduce transgene reactivation in HeLa cells   By contrast, the involvement of GCN2 and the ISR, including the downstream effectors ATF4 and CHOP, has never been tested  In addition, this pathway has been typically assessed in transient assays, lasting for a few hours, which may not be comparable with the prolonged starvation conditions necessary to reactivate the transgene expression (at least 15–24 h)  Thus, we tested whether CHOP expression was upregulated upon incubation of HeLa-OA1, HepG2-OA1 and C2C12-GFP cells in media deprived of different EAAs for 24–48 h 
As shown in Fig 3 and S4B Fig, we found that CHOP expression is increased in all EAA-starvation conditions, but not in the absence of Ala, in all tested cell lines  Similar, yet less pronounced, results were obtained with ATF4, consistent with the notion that activation of this transcription factor is mainly mediated by translational upregulation (not shown) [15, 26]  However, the upregulation of CHOP does not parallel quantitatively that of the transgene, neither appears sufficient to induce it  In fact, CHOP is highly upregulated even upon Trp starvation, which consistently results in no or minimal reactivation of the transgenes (compare CHOP with OA1 or GFP expression; Fig 3 and S4B Fig)  Thus, while the ISR appears widely activated upon EAA starvation, the upregulation of its downstream effector CHOP only partly correlates with transgene reactivation and may not be sufficient to induce it 
The activation of the ISR upon AA starvation suggests that GCN2 may be involved in the transgene reactivation response  Therefore, we tested whether direct pharmacological activation of this kinase is sufficient to trigger the transgene reactivation similarly to starvation  In addition, we used pharmacological inhibitors of mTOR to corroborate previous negative results in HeLa cells  in the other cell lines under study  To this aim, HeLa-OA1 or GFP, HepG2-OA1 and C2C12-GFP cells were cultured in the presence of different concentrations of PP242 (mTOR inhibitor) or L-Histidinol (GCN2 activator, inhibiting tRNAHis charging by histidyl-tRNA synthetase), either alone or in combination for 24 h, compared to Met/Cys-deprived and full medium  As shown in Fig 4 and S5 Fig, while inhibition of mTORC1 consistently leads to minor or no effects, in agreement with previous findings , treatment with L-Histidinol results in efficient reactivation of the transgene in HepG2-OA1 and C2C12-GFP cells, but not in HeLa cells 
Fig 4  mTOR inhibition and GCN2 activation differently affect transgene expression in HeLa and HepG2 cells 
Relative transgene (OA1) and CHOP mRNA abundance in HeLa-OA1 (A) and HepG2-OA1 (B) cells, cultured in Met/Cys-deprived medium, or in the presence of PP242 (mTOR inhibitor; 1–3 μM) or L-Histidinol (HisOH, GCN2 activator; 4–16 mM), either alone or in combination for 24–48 h, compared to full medium  Mean ± SEM of 4 (A) or 3 (B) independent experiments  Data are expressed as fold change vs  control (full medium = 1)  *P<0 05, **P<0 01, ***P<0 001 (one way ANOVA, followed by Dunnett’s post-test vs  full medium)  PP-1 and PP-3, PP242 at 1 and 3 μM, respectively; HisOH-4 and HisOH-16, L-Histidinol at 4 and 16 mM, respectively 
Specifically, L-Histidinol is not effective in HeLa-OA1 and HeLa-GFP cells, either alone or in combination with PP242 (Fig 4A and S5A Fig), or by using different concentrations of the drug, with or without serum (not shown)  In these cells, L-Histidinol appears also unable to trigger the ISR, as indicated by lack of CHOP upregulation, possibly due to their different sensitivity to the drug  These findings are consistent with previous reports, describing the use of L-Histidinol in HeLa cells in conditions of low His concentration in the culture medium , which would resemble AA starvation in our system and therefore may not be applicable  Thus, even though the amount of the amino alcohol was adapted to exceed 20 to 80 times that of the amino acid, as described , HeLa cells may be resistant or able to compensate 
In contrast, in other cell types, L-Histidinol has been utilized in regular DMEM, to mimic the AA response triggered by DMEM lacking His [48, 49]  Consistently, in HepG2-OA1 cells, L-Histidinol is sufficient to elicit extremely high levels of transgene reactivation, and its combination with PP242 results in additive or even synergistic effects, possibly due to an indirect effect of mTOR inhibition on GCN2 activity (Fig 4B) [50, 51]  Similarly, C2C12-GFP cells efficiently reactivate the transgene upon treatment with L-Histidinol, but not PP242 (S5B Fig)  However, differently from HepG2-OA1 cells, simultaneous treatment of C2C12-GFP cells with L-Histidinol and PP242 does not lead to synergistic effects  Consistent with stimulation of the ISR, CHOP and to a minor extent ATF4 are upregulated by L-Histidinol in both cell lines, yet their expression levels show only an incomplete correlation with those of the transgene (Fig 4B, S5B Fig, and not shown) 
The finding that GCN2 activation by L-Histidinol is sufficient to reactivate the transgenes in both HepG2-OA1 and C2C12-GFP cells pointed to this kinase, and to the downstream ISR, as the pathway possibly involved in the EAA starvation response  Thus, we investigated whether the ISR is sufficient to trigger upregulation of the OA1 transgene in HepG2-OA1 cells by pharmacological means  As CHOP expression does not correspond quantitatively and is not sufficient to induce transgene reactivation, we tested the role of the core upstream event of the ISR, namely the phosphorylation of eIF2α , which can be induced by pharmacological treatments, independent of GCN2 (Fig 5A)  To this aim, we used Salubrinal, a specific phosphatase inhibitor that blocks both constitutive and ER stress-induced phosphatase complexes against eIF2α, thereby increasing its phosphorylation   We found that, while the ISR is activated upon Salubrinal treatment, as shown by increased CHOP expression, it does not induce OA1 transgene reactivation (Fig 5B) 
Fig 5  The ISR is neither sufficient nor necessary to induce transgene reactivation in HepG2 cells 
(A) Schematic representation of GCN2 activation by AA starvation, resulting in phosphorylation of eIF2a and initiation of the downstream ISR  In addition to GCN2, the ISR may be activated by other eIF2a kinases (PKR, HRI and PERK; not shown in the picture)  (B) Relative transgene (OA1) and CHOP mRNA abundance in HepG2-OA1 cells treated for 24 h with Salubrinal (a drug that induces the ISR by inhibiting the dephosphorylation of eIF2α; 75 μM), compared to full medium  Mean ± range of two experiments  Data are expressed as fold change vs  control (DMEM = 1)  *P<0 05 (paired two-tailed Student’s t-test vs  control)  (C) Relative transgene (OA1) and CHOP mRNA abundance in HepG2-OA1 cells treated for 6 h with L-Histidinol (HisOH, GCN2 activator; 4 mM), in the absence or presence of ISRIB (a drug that bypasses the phosphorylation of eIF2α, inhibiting triggering of the ISR; 100 nM)  Mean ± range of two experiments  Data are expressed as fold change vs  control (DMEM = 1)  **P<0 01, ***P<0 001 (one way ANOVA, followed by Tukey’s post-test; P values refer to comparisons vs  control, unless otherwise indicated)  (D) Relative transgene (OA1) and ATF4 mRNA abundance in HepG2-OA1 cells transfected with control (CTRL) or anti-ATF4 siRNAs, and incubated in the presence or absence of L-Histidinol (HisOH, GCN2 activator; 4 mM) for 6 h  Mean ± range of two experiments  Data are expressed as fold change vs  control (w/o HisOH = 1, top; control siRNA = 1, bottom)  *P<0 05 (one way ANOVA, followed by Tukey’s post-test; P values refer to comparisons vs  control, unless otherwise indicated) 
To test whether the ISR is necessary to trigger the transgene response to L-Histidinol, we used the chemical compound ISRIB, which inhibits the activation of the ISR, even in the presence of phosphorylated eIF2α, likely by boosting the activity of the guanine-nucleotide exchange factor (GEF) for eIF2α, namely eIF2B [53, 54]  HepG2-OA1 cells were stimulated with L-Histidinol, either in the presence or absence of ISRIB  As shown in Fig 5C, while the expression of CHOP is inhibited by ISRIB, as expected, the reactivation of the OA1 transgene is not affected  In addition, knockdown of the closest eIF2α downstream effector ATF4 by siRNAs does not interfere with the reactivation of the OA1 transgene by L-Histidinol (Fig 5D)  Together, these data suggest that eIF2α phosphorylation and the downstream ISR pathway are neither sufficient nor necessary to induce transgene reactivation 
To definitively establish if GCN2 is necessary to trigger the transgene reactivation response to EAA starvation, we directly suppressed its expression by CRISPR/Cas9-mediated knock-out (KO)  We generated three independent GCN2-KO clones from the parental HeLa-OA1 cell line, by using three different guide RNAs, two against exon 1 (clones 183#11 and 185#5), and one against exon 6 (clone 239#1) of the GCN2 gene  Genomic characterization confirmed the presence of mutations on both alleles of exon 1 of the GCN2 gene in clone 183#11, and on both alleles of exon 6 in clone 239#1; by contrast, clone 185#5 showed multiple alleles in exon 1, consistent with the presence of two cell populations, and was not characterized further at the genomic level (S6 Fig)  None of these clones express GCN2 at the protein level, as shown by immunoblotting (Fig 6A)  To test the GCN2-KO cells for their ability to respond to EAA starvation, parental HeLa-OA1 cells and the three GCN2-KO clones were cultured in media deprived of Met/Cys or Thr (corresponding to the most effective treatments in this cell line; see Fig 3A) for 24–48 h and transgene expression was assessed by RT-qPCR  We found that the reactivation of the OA1 transgene is neither abolished, nor reduced by KO of GCN2, thus excluding that this kinase is necessary for the response to EAA starvation in HeLa-OA1 cells (Fig 6B and 6C) 
Fig 6  GCN2 knockout does not interfere with transgene reactivation in HeLa cells 
(A) Immunoblotting of protein extracts from the HeLa-OA1 parental cell line and GCN2-KO clones 183#11, 185#5 and 239#1, immunodecorated with anti-GCN2 antibody  Arrow, GCN2 specific band  Ponceau staining was used as loading control  (B, C) Relative transgene (OA1) mRNA abundance in HeLa-OA1 cells and GCN2-KO clones, cultured in Met/Cys (B) or Thr (C) deprived medium for 24 h or 48 h, respectively, compared to full medium  Mean ± SD of 3 technical replicates from 1 experiment  Data are expressed as fold change vs  control (full medium = 1)  Since independent clones may display variable reactivation responses (e g  due to different levels of transgene expression in basal conditions), the results are not shown as means of the three clones, but as separate replicates 
Similarly, we generated GCN2-KO clones from the parental HepG2-OA1 cell line by the same strategy  By using a guide RNA against exon 1 of the GCN2 gene, we obtained three independent GCN2-KO clones, namely E23, F22 and F27  Genomic characterization confirmed the presence of mutations on both alleles of exon 1 of the GCN2 gene in clone F27 (S7 Fig) and all three clones showed a very low amount—if any—of residual GCN2 protein, compared to the original HepG2-OA1 cell line (Fig 7A)  To assess the ability of GCN2-KO cells to reactivate the transgene upon starvation, we cultured parental HepG2-OA1 cells and the three GCN2-KO clones in media deprived of Met/Cys or His (corresponding to the most effective treatments in this cell line; see Fig 3B) for 24 h, and evaluated the transgene expression by RT-qPCR  As shown in Fig 7B and 7C, we found that the reactivation of the OA1 transgene is neither abolished, nor reduced by KO of GCN2, as in HeLa cells  To further confirm this result, we knocked-down GCN2 by RNA interference (RNAi), and incubated the cells with or without L-Histidinol for 6 h  As shown in Fig 8, treatment of HepG2-OA1 cells with L-Histidinol results in efficient transgene reactivation, even upon significant GCN2 downregulation, both at the mRNA and protein levels  Taken together, these data strongly support the conclusion that GCN2 is not necessary for transgene reactivation in response to EAA starvation, either in HeLa or in HepG2 cells 
Fig 7  GCN2 knockout does not interfere with transgene reactivation in HepG2 cells 
(A) Immunoblotting of protein extracts from the HepG2-OA1 parental cell line and GCN2-KO clones 185#27, E23, F22, F27, immunodecorated with anti-GCN2 antibody  Clone 185#27 results from the first round of selection, and was used to generate clones E23, F22, F27  Arrow, GCN2 specific band  For GCN2 protein quantification, Ponceau staining was used as loading control and data are expressed as fold change vs  parental cell line (= 1)  (B, C) Relative transgene (OA1) mRNA abundance in HepG2-OA1 cells and GCN2-KO clones, cultured in Met/Cys (B) or His (C) deprived medium for 24 h, compared to full medium  Mean ± SD of 3 technical replicates from 1 experiment 



ground_truth: Using the vacuum Einstein equation and the Baez-Bunn form.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Is the ISR necessary for transgene reactivation?

Read the following text and answer briefly.

Context 3: Paper Info

Title: Compressed quantum error mitigation
Publish Date: 10 May 2023
Author List: Maurits Tepaske (from Physikalisches Institut, Universität Bonn), David Luitz (from Physikalisches Institut, Universität Bonn)

Figure

FIG 3 The out-of-time-ordered correlator C otoc i=L/2,j (t) as a function of the operator position j and time t, for the infinite temperature initial state, for a denoised second-order Trotter supercircuit with Trotter depth Mtrot = 32 and denoiser depth M = 2 We consider evolution times t = 0 5, 1,    , 5, for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarizing noise with p = 0 01 
FIG  4  The complex eigenvalues λ of the noisy second-order Trotter supercircuit with Mtrot = 16 at time t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised Trotter supercircuit (right) The Trotter circuit is for a L = 6 Heisenberg model with PBC, and all twoqubit channels are affected by depolarizing noise with p = 0 0046 The unit circle, on which unitary eigenvalues must lie, is shown in black, and the noiseless eigenvalues are shown as blue bars It is evident that the denoiser recovers all the noiseless eigenvalues from the noisy circuit 
FIG  2  The complex eigenvalues λ of the noisy second-order Trotter supercircuit with Mtrot = 16 at time t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised Trotter supercircuit (right) The Trotter circuit is for a L = 6 Heisenberg model with PBC, and all twoqubit channels are affected by depolarizing noise with p = 0 036 The unit circle, on which unitary eigenvalues must lie, is shown in black, and the noiseless eigenvalues are shown as blue bars It is clear that the denoiser recovers with high accuracy the noiseless eigenvalues from the noisy circuit 
FIG  3  The half-chain channel entanglement entropy S at different two-qubit depolarizing noise strengths p, for a secondorder Trotter supercircuit with Mtrot = 16 and t = 2, for a M = 4 denoiser The Trotter circuit is for a Heisenberg model with PBC of size L = 6 The different curves correspond to the different supercircuits, i e  the noisy supercircuit, the denoiser, the corresponding denoised supercircuit, and the noiseless variant 
FIG  4  The out-of-time-ordered correlator C otoc i=L/2,j (t) as a function of the operator position j and stacked time t, for the infinite temperature initial state, for a denoised secondorder Trotter supercircuit with Trotter depth Mtrot = 32 and denoiser depth M = 2 It is optimized at t = 2 and stacked up to ten times The calculations are for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarization with p = 0 01 The denoiser is affected by the same noise 
FIG 6 The distribution of the ZZ angle α of M = 2 denoisers (top panels) and M = 8 denoisers (bottom panels), with the lightest color corresponding to the denoiser for the Trotter supercircuit with t = 0 5, and the darkest color with t = 5 As usual, we consider the Heisenberg model on a periodic chain, and second-order Trotter supercircuits with depths Mtrot = 8, 16, 32, 64, which together with the denoiser is affected by a two-qubit depolarizing noise with p = 0 01 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively 
FIG  7  The sampling overhead γ of the optimized denoisers from Fig  2 of the main text, with denoiser depths M = 1, 2, 4, 6, 8 and Trotter depths Mtrot = 8, 16, 32, 64 at times t = 0 5, 1,    , 5, for the Heisenberg model on a chain with PBC affected by two-qubit depolarizing noise with p = 0 01 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively 
FIG 8 The domain wall magnetization Z dw after evolving a periodic density wall |dw |dw * with the denoised second-order Trotter supercircuits D C from Fig 2of the main text These supercircuits have various Trotter depths Mtrot = 8, 16, 32, 64, denoiser depths M = 1, 2, 4, 6, 8, and evolution times t = 0 5, 1,    , 5, for the periodic L = 14 Heisenberg chain that is affected by two-qubit depolarizing noise of strength p = 0 01 The denoiser is affected by the same noise The non-denoised results are labelled with M = 0 and the noiseless results with p = 0 The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively We see that the denoiser allows us to recover the noiseless behavior 

abstract

We introduce a quantum error mitigation technique based on probabilistic error cancellation to eliminate errors which have accumulated during the application of a quantum circuit  Our approach is based on applying an optimal "denoiser" after the action of a noisy circuit and can be performed with an arbitrary number of extra gates 
The denoiser is given by an ensemble of circuits distributed with a quasiprobability distribution  For a simple noise model, we show that efficient, local denoisers can be found, and we demonstrate their effectiveness for the digital quantum simulation of the time evolution of simple spin chains  Introduction 
-Quantum information processing has been theoretically shown to hold great promises, and quantum algorithms were developed which can in principle achieve an exponential speed-up over their classical counterparts, both for general purpose computing and quantum simulation   However, present day quantum computing prototypes still suffer from significant noise processes which hinder the execution of many potentially groundbreaking quantum algorithms  
Nontrivial quantum algorithms typically require large sequences of quantum gates, each of which introduces dissipation and hence an overall loss of coherence, eventually rendering the results useless  Until quantum error correction becomes practical, quantum error mitigation seems to be more feasible to increase the accuracy of expectation values 
Here the goal is to induce the (partial) cancellation of errors that stem from noisy quantum gates by extending the circuit corresponding to the desired algorithm with an ensemble of gates , sampled from a quasiprobability distribution  The traditional way to accomplish this is with the gatewise method from , where noise is mitigated by inverting the noise channel of each gate separately, i e  the cancellation of errors is performed for each gate on its own 
Here the local noise channel is approximated in a way such that it can be easily inverted analytically, e g  using Pauli twirling   Gates are then sampled from the inverted noise channel by interpreting it as a quasiprobability distribution  Because in this gate-wise approach every noisy gate has to be modified separately, the sign problem is exponentially large in the number of gates, limiting the practicality of the mitigation 
The success of the gate-wise approach resulted in a large body of work concerning these methods , including extensions for simultaneous mitigation of multiple gates by Pauli-twirling entire layers or variationally constructing a mitigating matrix product operator   In principle, errors during the execution of a circuit can propagate and accumulate 
These propagated errors * david luitz@uni-bonn de ≈ C

C

FIG  1  An example of the quantum error mitigation procedure used in this work for the time evolution of the wave function of a spin chain  The ideal second-order Trotter supercircuit C of depth Mtrot = 1 (light blue) is approximated by applying a denoiser D of depth M = 1 (red) to the noisy Trotter supercircuit C (dark blue) 
Because the denoiser is applied after fully executing the noisy Trotter supercircuit, it represents an approximate inverse of the global noise channel with a precision tunable by the depth of the denoiser  can potentially blow up and lead to large errors for the circuit as a whole   Here we introduce a mitigation technique that takes into account the propagation of errors, can be performed with a tunable number of extra gates, and works for non-Clifford local noise channels since the inversion of the accumulated global noise channel is implicit 
We first execute the targeted noisy circuit completely, letting the noise propagate and accumulate, and only afterwards we apply an extra random circuit sampled from a quasiprobability distribution  We call the corresponding ensemble of random circuits a denoiser, and we construct it such that upon averaging the accumulated errors cancel 
Essentially, the denoiser inverts a global noise channel  Since we will construct it as a local brickwall circuit, following the classical preprocessing approach from , we call this compressed quantum error mitigation  Method  -Due to the inevitable coupling of a quantum processor to its environment, every qubit operation is affected by noise 
Therefore, the simplest technique to minimize the impact of the resulting noise is to minimize the number of operations when performing a quantum algorithm  In we showed that many-body time evolution operators can be efficiently compressed into brick-wall circuits with high fidelity per gate  In this Letter, we consider the noise explicitly by treating quantum operations as (generally non-unitary) quantum channels, corresponding to completely positive and trace preserving (CPTP) maps  
For example, instead of a noiseless two-qubit gate G, which acts on a quantum state |ρ in superoperator form as G|ρ = G⊗G * |ρ , we get the noisy channel G = N G, where the noise channel N implements the two-qubit noise   These channels are used to construct a "supercircuit" C = N G i=1 Gi , consisting of N G channels, which is affected by multi-qubit accumulated noise 
This supercircuit encodes an ensemble of circuits   For simplicity, we assume that the noisy channels Gi in each half brickwall layer are lattice inversion and translation invariant, such that we can construct a denoiser with these properties, limiting the number of variational parameters  The purpose of quantum error mitigation is to modify the ensemble of circuits described by C in a way that we can use it to obtain the noiseless expectation values 
In superoperator language, we do this by following the supercircuit C with a denoiser supercircuit D, such that D C is as close to the noiseless supercircuit C = C ⊗ C * as possible  Here C is the target unitary circuit  Because the noise channel N is non-unitary, hence making the supercircuit C non-unitary, we need to use a non-unitary denoiser to retrieve the unitary C 
We illustrate the mitigation procedure in Fig  , where a denoiser with one layer is used to mitigate errors for a second-order Trotter supercircuit with one layer  This circuit architecture is commonly used to simulate the time evolution of a quantum many-body system, until some time t, with controllable precision , and we will use it to benchmark the denoiser 
In practice, we cannot directly implement a supercircuit, and so we have to utilize its interpretation as an ensemble of circuits  Essentially, after executing a shot of the noisy circuit we sample the denoiser and apply it  The goal is to construct the denoiser in a way that averaging over many of its samples cancels the accumulated errors and gives us a good approximation of the noiseless expectation values 
It should be noted that our approach requires more gate applications on the quantum processor than with the gate-wise scheme, since there each sample from the mitigation quasiprobability distribution can be absorbed into the original circuit, whereas our approach increases the circuit depth  We take this into account by imposing the same noise on the denoiser 
Furthermore, within our scheme, the dimensionality of the quasiprobabilistic mitigating ensemble can be controlled, in contrast to the gate-wise approach where it is equal to the gate count  To facilitate the stochastic interpretation we parameterize each two-qubit denoiser channel G i as a sum of CPTP maps, such that we can sample the terms in this sum and execute the sampled gate on the quantum processor 
Concretely, we use a trace preserv-ing sum of a unitary and a non-unitary channel  For the unitary part we take a two-qubit unitary channel U( φ i ) = U ( φ i ) ⊗ U * ( φ i ), with U ( φ i ) a two-qubit unitary gate parameterized by φ i   For this we take the two-qubit ZZ rotation exp(−iα(σ z ⊗ σ z )) with angle α, which can be obtained from native gates on current hardware , and dress it with four general one-qubit unitaries, only two of which are independent if we want a circuit that is space inversion symmetric around every bond 
The resulting gate has 7 real parameters φ i   For the non-unitary part, which is essential because D has to cancel the non-unitary accumulated noise to obtain the noiseless unitary circuit, we use a general onequbit measurement followed by conditional preparation channel M( , with V a general one-qubit unitary and each κ i a 3-dimensional vector, resulting in a real 9-dimensional ζ i  
This yields the two-qubit correlated measurement M( With these parts we construct the parameterization with coefficients η i ∈ R that satisfy η 0 + η 1 = 1 because G i is trace preserving  Note that here the tensor product symbol corresponds to combining two one-qubit channels to make a two-qubit channel, whereas in most of the paper it is used to link the column and row indices of a density matrix 
We construct the denoiser from the noisy channels Gi = N G i   With this parameterization one denoiser channel has 17 independent real parameters, such that a denoiser of depth M , i e  consisting of M brickwall layers, has 34M real parameters (we use one unique channel per half brickwall layer)  For reference, a general channel has 544M parameters 
To determine the mitigated expectation values we use the full expression where |ρ 0 is the initial state and |1 is the vectorized identity operator on the full Hilbert space  To evaluate this on a quantum processor, we use the stochastic interpretation of (1) to resample   In particular, from each channel (1) we get a unitary with probability p 0 = |η 0 |/γ and a measurement followed by conditional preparation with probability p 1 = |η 1 |/γ 
Here γ = |η 0 | + |η 1 | is the sampling overhead, which characterizes the magnitude of the sign problem from negative η i   For quasiprobability distributions, i e  with γ > 1, every denoiser sample has an extra sign sgn(η) = N G g=1 sgn(η g ), 2  The normalized distance between the denoised Trotter supercircuit D C and the noiseless Trotter supercircuit C (top panels), at evolution times t = 0 5, 1,    , 5, and the twopoint z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t (bottom panels), for the infinite temperature initial state 
We consider denoisers with depths M = 1, 2, 4, 6, 8 and second-order Trotter circuits with depths Mtrot = 16, 32, 64  In the top panels we use a Heisenberg chain with L = 8, and in the bottom panels with L = 14, both with periodic boundary conditions  All gates are affected by two-qubit depolarizing noise with p = 0 01 
The non-denoised results are labelled with M = 0, and the noiseless values with p = 0  where sgn(η g ) is the sign of the sampled coefficient of the gth channel  γ = 1 means that all signs are positive  Observables Ô p=0 for the noiseless circuit are then approximated by resampling the observables from the denoiser ensemble
where γ = N G g=1 γ g is the overall sampling overhead, with γ g the overhead of the gth gate  Clearly, a large γ implies a large variance of Ô p=0 for a given number of samples, with accurate estimation requiring the cancellation of large signed terms  The number of samples required to resolve this cancellation of signs is bounded by Hoeffding's inequality, which states that a sufficient number of samples to estimate Ô p=0 with error δ at probability 1 − ω is bounded by (2γ 2 /δ 2 ) ln(2/ω)  
Since γ scales exponentially in γ g , it is clear that a denoiser with large M and γ 1 will require many samples  We observed that decompositions with γ > 1 are crucial for an accurate denoiser  Restricting to γ = 1 leads to large infidelity and no improvement upon increasing the number of terms in or the depth M of the denoiser 
Simply put, probabilistic error cancellation of gate noise introduces a sign problem and it is crucial to find optimal parameterizations (1) which minimize γ to make the approach scalable  This issue arises in all high performance error mitigation schemes , because the inverse of a physical noise channel is unphysical and cannot be represented as a positive sum over CPTP maps 
This is clearly visible in the spectra of the denoiser, which lies outside the unit circle (cf  Fig  )  This makes the tunability of the number of gates in each denoiser sample a crucial ingredient, which allows control over the sign problem, because we can freely choose the η i in   For the parametrization (1) of denoiser channels, we try to find a set of parameters for error mitigation by minimizing the normalized Frobenius distance between the noiseless and denoised supercircuits
which bounds the distance of output density matrices and becomes zero for perfect denoising  We carry out the minimization of on a classical processor, using gradient descent with the differential programming algorithm from   Instead of explicitly calculating the accumulated global noise channel and subsequently inverting it, we approximate the noiseless supercircuit C with the denoised supercircuit D C, effectively yielding a circuit representation D of the inverse noise channel 
Results  -To benchmark the denoiser we apply it to the second-order Trotter circuits of the spin-1/2 Heisenberg chain with periodic boundary conditions (PBC) where is the Pauli algebra acting on the local Hilbert space of site i  A second-order Trotter circuit for evolution time t with depth M trot consists of M trot − 1 half brickwall layers with time step t/M trot and two layers with half time step  
We consider circuits that are affected by uniform depolarizing noise with probability p for simplicity, but our approach can be used for any non-Clifford noise  The two-qubit noise channel is which acts on neighboring qubits i and i + 1 and is applied to each Trotter and denoiser gate, and p = 0 01 unless stated otherwise 
We study circuits with depths M trot = 16, 32, 64 for evolution times t = 0 5, 1,    , 5, and denoisers D with depths M = 1, 2, 4, 6, 8  In the top panels of Fig  we show (4) for a chain of size L = 8 as a function of time t  Here it can be seen that even for M trot = 32 a denoiser with M = 1 already improves by roughly an order of magnitude at all considered t 
Depending on M trot and t, further increasing M lowers , with the biggest improvements occurring for high precision Trotter circuits with large depth M trot = 64 and short time t = 0 5, where the Trotter gates are closer to the identity than in the other cases  At the other extreme, for M trot = 16 the improvements are relatively small upon increasing M > 2  In all cases the denoiser works better at early times than at late times, again indicating that it is easier to denoise Trotter gates that are relatively close to the identity 
To probe the accuracy of the denoiser on quantities that do not enter the optimization, as a first test we consider the two-point correlator between spins at different times where we have chosen the infinite temperature initial state, and C(t) is the Trotter supercircuit for time t  In the bottom panels of Fig  we show C zz i=L/2,j=L/2 (t) for the supercircuits from the upper panels, now for a L = 14 chain 
Here we see that at M trot = 16 we can retrieve the noiseless values already with M = 1, but that increasing M trot makes this more difficult  At M trot = 64 we see larger deviations, and improvement upon increasing M is less stable, but nonetheless we are able to mitigate errors to a large extent  As a further test, we compute the out-of-time-ordered correlator (OTOC) ]
In Fig  we show the results for i = L/2, for a Trotter circuit with depth M trot = 32 and a denoiser with depth M = 2  Here we see that a denoiser with M M trot is able to recover the light-cone of correlations, which are otherwise buried by the noise  In the Supplementary Material we consider how the denoiser performs at different noise levels p, and how the denoised supercircuits perform under stacking 
There we also calculate domain wall magnetization dynamics, and show the distribution of the optimized denoiser parameters and the sampling overhead associated to the denoiser as a whole  In Fig  we show the eigenvalues of the noisy supercircuits for a noisy second-order Trotter supercircuit with M trot = 16 at t = 1 (left), the corresponding optimized denoiser with M = 4 (center), and the denoised supercircuit (right) 
The eigenvalues λ of a unitary supercircuit lie on the unit circle, and in the presence of dissipation they are pushed to the center  We see that the spectrum of the denoiser lies outside the unit circle, making it an unphysical channel which cures the effect of the noise on the circuit, such that the spectrum of the denoised circuit is pushed back to the unit circle 
The noiseless eigenvalues are shown as blue bars, making it clear that the denoiser is able to recover the noiseless eigenvalues from the noisy circuit  In the Supplementary Material we show the spectra for a p = 0 036 denoiser, where we observe a clustering of eigenvalues reminiscent of Refs    There we also investigate the channel entropy of the various supercircuits  
Conclusion  -We have introduced a probabilistic error cancellation scheme, where a classically determined denoiser mitigates the accumulated noise of a (generally non-Clifford) local noise channel  The required number of mitigation gates, i e  the dimensionality of the corresponding quasiprobability distribution, is tunable and the parameterization of the corresponding channels provides control over the sign problem that is inherent to probabilistic error cancellation 
We have shown that a denoiser with one layer can already significantly mitigate errors for second-order Trotter circuits with up to 64 layers  This effectiveness of low-depth compressed circuits for denoising, in contrast with the noiseless time evolution operator compression from , can be understood from the non-unitarity of the denoiser channels 
In particu-lar, measurements can have non-local effects, since the measurement of a single qubit can reduce some highly entangled state (e g  a GHZ state) to a product state, whereas in unitary circuits the spreading of correlations forms a light-cone  To optimize a denoiser with convenience at L > 8, the optimization can be formulated in terms of matrix product operators or channels , which is convenient because the circuit calculations leading to the normalized distance and its gradient are easily formulated in terms of tensor contractions and singular value decompositions  
This provides one route to a practical denoiser, which is relevant because the targeted noiseless circuit and the accompanying noisy variant in (4) need to be simulated classically, confining the optimization procedure to limited system sizes with an exact treatment or limited entanglement with tensor networks 
Nonetheless, we can use e g  matrix product operators to calculate (4) for some relatively small t, such that the noiseless and denoised supercircuits in (4) have relatively small entanglement, and then stack the final denoised supercircuit on a quantum processor to generate classically intractable states 
Analogously, we can optimize the channels exactly at some classically tractable size and then execute them on a quantum processor with larger size  Both approaches are limited by the light-cone of many-body correlations, as visualized in Fig  , because finite-size effects appear when the light-cone width becomes comparable with system size 
1  The normalized distance (left) and z spin correlator C zz i=L/2,j=L/2 (right), for a second-order Trotter supercircuit of depth Mtrot = 16 for time t = 1, affected by various twoqubit depolarizing errors p  We compare the values obtained with and without a denoiser, i e  M > 0 and M = 0, to the noiseless values (p = 0) 
The denoiser is affected by the same noise as the Trotter circuit  We consider denoisers with depths M = 1, 2, 4, 6, 8, and we use a L = 8 Heisenberg chain with PBC for the normalized distance, while for the correlator we use L = 14  * david luitz@uni-bonn de to observe that even for larger noise strength p, the local observable C zz improves significantly even with denoisers of depth M = 1 
For large noise strengths, we generally see that the optimization of the denoiser becomes difficult, leading to nonmonotonic behavior as a function of p, presumably because we do not find the global optimum of the denoiser  It is interesting to analyze the spectra of the supercircuits considered in this work 
As mentioned in the main text, the spectrum of the ideal, unitary supercircuit C lies on the unit circle  The comparison to this case is therefore instructive  In the main text, we showed an example of the spectra in Fig  for moderate noise strength  Here, we show additional data for stronger noise p = 0 036 in Fig  for a denoiser with M = 4 layers, optimized to mitigate errors for a second-order Trotter supercircuit with M trot = 16 layers at time t = 1 
The eigenvalues λ of the noisy supercircuit C are clustered close to zero, far away from the unit circle (except for λ = 1), showing that the circuit is strongly affected by the noise  To mitigate the impact of the noise, the denoiser consequently has to renormalize the spectrum strongly  If it accurately represents the inverse of the global noise channel, its spectrum has to lie far outside the unit circle, which is the case 
Interestingly, we observe a clustering of eigenvalues which is reminiscent to the spectra found in   By comparison to these works, we suspect that this is due to the local nature of the denoiser, and warrants further investigation  The right panel of Fig  shows the result of the denoiser, pushing the eigenvalues back to the unit circle, nearly with the exact same distribution along the circle as the noiseless eigenvalues (blue bars) 
Due to the strong noise, this is not achieved perfectly, and it is clear that this cannot work in principle if the global noise channel has a zero eigenvalue  The complexity of an operator can be quantified by its operator entanglement entropy   Here we calculate the half-chain channel entanglement entropy S of the noiseless C, noisy C, denoiser D, and denoised D C supercircuits 
We define S as the entanglement entropy of the state that is related to a supercircuit C via the Choi-Jamio lkowski isomorphism, i e  ψ C = χ C /N , where the process matrix χ ab,cd C = C ac,bd is simply a reshaped supercircuit and N ensures normalization  Then we have S = −Tr [ψ C ln ψ C ]  This entropy measure is a particular instance of the "exchange entropy", which characterizes the information exchange between a quantum system and its environment  
In Fig  we plot the various S for a second-order Trotter circuit with M trot = 16 at t = 2, for a denoiser with M = 4, both affected by two-qubit depolarizing noise with p ∈ [10 −3 , 10 −1 ]  The Trotter circuit is for a Heisenberg model with L = 6 and PBC  We see that at large p, the noise destroys entanglement in the noisy supercircuit, and that the denoiser S increases to correct for this, such that the denoised supercircuit recovers the noiseless S 
Here we investigate how denoised supercircuits perform upon repeated application  We optimize the denoiser for a Trotter supercircuit for a fixed evolution time t  Then, to reach later times, we stack the denoised supercircuit n times to approximate the evolution up to time nt: In Fig  we stack a denoised t = 1 supercircuit up to n = 20 times and calculate the correlation function, defined in the main text, for the middle site 
We consider Trotter depths M trot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8, for a L = 14 Heisenberg chain with p = 0 01 depolarizing two-qubit noise  The noisy results correspond to M = 0 and the noiseless results to p = 0  In Fig  we calculate the OTOC, defined in the main text, with stacked time evolution for a denoised t = 2 supercircuit with M trot = 32 and M = 2, stacked up to ten times 
We see that the stacked supercircuit performs very well, and the additional precision obtained by using deep denoisers (M = 8) pays off for long evolution times, where we see convergence to the exact result (black dashed lines in Fig  ) as a function of M   FIG    The two-point z-spin correlator C zz i=L/2,j=L/2 (t) of a spin on the middle site at times 0 and t, for the infinite temperature initial state, for denoised second-order Trotter supercircuits that are optimized at evolution time t = 1 and then stacked up to twenty times 
We use Trotter depths Mtrot = 8, 16, 32, 64 and denoiser depths M = 1, 2, 4, 6, 8  The calculations were performed for a periodic Heisenberg model with L = 14 and PBC, affected by two-qubit depolarizing noise with strength p = 0 01, which also affects the denoiser  The non-denoised results are labelled with M = 0, and the noiseless results with p = 0 
The panels are arranged as Mtrot = 8, 16, 32, 64 for top left, top right, bottom left, bottom right, respectively  The costliest and most noise-susceptible operation is the two-qubit ZZ rotation with angle α, which is the foundation of the unitary piece in our channel parameterization, defined in the main text 
For completeness, we here present the α angles of the optimized denoisers  The results are shown in Fig  , which contains histograms for the channel count N G versus α  The histograms are stacked, with the lightest color corresponding to the angles of the denoiser at t = 0 5 and the darkest at t = 5  The top four panels are for a denoiser with M = 2 and the bottom four with M = 8 
We consider M trot = 8, 16, 32, 64  We see that in both cases the distribution widens upon increasing M trot , indicating that the unitary channels start deviating more from the identity  Moreover, while the M = 2 denoisers in all cases except M trot = 64 have ZZ contributions close to the identity, this is clearly not the case for M = 8 
For simplicity, we did not focus on obtaining denoisers with the smallest sampling overhead γ, which is required to minimize the sign problem and hence ease the sampling of mitigated quantities  Instead, we let the optimization freely choose the η i in the denoiser parameterization, as defined in the main text 
In Fig  we show the sampling overhead of the denoisers from Fig  of the main text  We see that for M = 1 and M = 2 the sampling overhead is relatively small and uniform across the different t, whereas for M > 2 the optimization sometimes yields a denoiser with large γ and other times with small γ  This could be related to the difference in α distributions from Fig   
The large fluctuations of γ appears to stem from the difficulty in finding optimal deep denoisers, and our optimization procedure likely only finds a local minimum in these cases  Here C(t) is the Trotter supercircuit for time t  In Fig  we show Z dw for the circuits from Fig 



ground_truth: No, it is not necessary.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Can the denoiser be applied to circuits with non-Clifford noise?

Read the following text and answer briefly.

Context 4: 32% Asian, 0 06% Pacific Islander, 0 79% from other races, and 1 16% from two or more races   1 94% of the population were Hispanic or Latino of any race  37 1% were of German, 12 9% Swedish, 12 1% American, 6 7% English and 6 3% Irish ancestry according to Census 2000 

There were 11,205 households, out of which 33 00% had children under the age of 18 living with them, 62 50% were married couples living together, 6 00% had a female householder with no husband present, and 28 90% were non-families  25 50% of all households were made up of individuals, and 11 80% had someone living alone who was 65 years of age or older   The average household size was 2 49 and the average family size was 2 99 

In the county, the population was spread out, with 25 40% under the age of 18, 10 30% from 18 to 24, 25 20% from 25 to 44, 21 80% from 45 to 64, and 17 30% who were 65 years of age or older   The median age was 38 years  For every 100 females there were 95 90 males   For every 100 females age 18 and over, there were 92 90 males 

The median income for a household in the county was $41,138, and the median income for a family was $48,243  Males had a median income of $33,530 versus $21,175 for females  The per capita income for the county was $18,921   About 4 20% of families and 6 60% of the population were below the poverty line, including 5 20% of those under age 18 and 8 10% of those age 65 or over 

Government

Presidential elections
McPherson county is often carried by Republican candidates  The last time a Democratic candidate has carried this county was in 1964 by Lyndon B  Johnson 

Laws
Following amendment to the Kansas Constitution in 1986, the county remained a prohibition, or "dry", county until 1996, when voters approved the sale of alcoholic liquor by the individual drink with a 30 percent food sales requirement 

Education

Colleges
 McPherson College in McPherson
 Bethany College in Lindsborg
 Central Christian College in McPherson

Unified school districts
 Smoky Valley USD 400
 McPherson USD 418
 Canton-Galva USD 419
 Moundridge USD 423
 Inman USD 448

School district office in neighboring county
 Goessel USD 411
 Little River-Windom USD 444

Museums
 Birger Sandzén Memorial Gallery in Lindsborg
 McCormick-Deering Days Museum in Inman
 McPherson Museum in McPherson
 Lindsborg Old Mill & Swedish Heritage Museum in Lindsborg
 Kansas Motorcycle Museum in Marquette

Communities

Cities

 Canton
 Galva
 Inman
 Lindsborg
 Marquette
 McPherson (county seat) 
 Moundridge
 Windom

Unincorporated communities
† means a Census-Designated Place (CDP) by the United States Census Bureau 
 Conway
 Elyria†
 Groveland
 Johnstown
 New Gottland
 Roxbury†

Ghost towns
 Alta Mills
 Battle Hill
 Christian
 Doles Park
 Elivon
 King City
 Sweadal

Townships
McPherson County is divided into twenty-five townships   The cities of Lindsborg and McPherson are considered governmentally independent and are excluded from the census figures for the townships   In the following table, the population center is the largest city (or cities) included in that township's population total, if it is of a significant size 

See also
 List of people from McPherson County, Kansas
 National Register of Historic Places listings in McPherson County, Kansas
 McPherson Valley Wetlands
 Maxwell Wildlife Refuge

References

Notes

Further reading

 Wheeler, Wayne Leland  "An Analysis of Social Change in a Swedish-Immigrant Community: The Case of Lindsborg, Kansas " (PhD dissertation, University of Missouri-Columbia; ProQuest Dissertations Publishing, 1959  5905657) 

County
 Through the Years: A Pictorial History of McPherson County; McPherson Sentinel' Heritage House Publishing Co; 1992 
 McPherson County First Courthouse Built About 1869 or 1870; Lindsborg News-Record; March 30, 1959 
 Pioneer Life and Lore of McPherson County, Kansas; Edna Nyquist; Democratic-Opinion Press; 1932 
 A History of the Church of the Brethren in Kansas (includes McPherson College history); Elmer LeRoy Craik; McPherson Daily; Republican Press; 397 pages; 1922 
 Portrait and Biographical Record of Dickinson, Saline, McPherson, and Marion Counties, Kansas; Chapman Bros; 614 pages; 1893 
 Standard Atlas of McPherson County, Kansas; Geo  A  Ogle & Co; 82 pages; 1921 
 Plat Book of McPherson County, Kansas; North West Publishing Co; 50 pages; 1903 
 Edwards' Atlas of McPherson County, Kansas; John P  Edwards; 51 pages; 1884 

Trails
 The Story of the Marking of the Santa Fe Trail by the Daughters of the American Revolution in Kansas and the State of Kansas; Almira Cordry; Crane Co; 164 pages; 1915  (Download 4MB PDF eBook)
 The National Old Trails Road To Southern California, Part 1 (LA to KC); Automobile Club Of Southern California; 64 pages; 1916  (Download 6 8MB PDF eBook)

Mennonite Settlements
 Impact of Mennonite settlement on the cultural landscape of Kansas; Brenda Martin; Kansas State University; 1985/1988  
 Mennonite settlement : the relationship between the physical and cultural environment; Susan Movle; University of Utah; 1975/1886 
 Status of Mennonite women in Kansas in their church and home relationships; Eva Harshbarger; Bluffton College; 1925/1945 

External links

County
 
 McPherson County - Directory of Public Officials
Historical
 , from Hatteberg's People'' on KAKE TV news
Maps
 McPherson County Maps: Current, Historic, KDOT
 Kansas Highway Maps: Current, Historic, KDOT
 Kansas Railroad Maps: Current, 1996, 1915, KDOT and Kansas Historical Society

 
Kansas counties
1867 establishments in Kansas
Populated places established in 1867



ground_truth: Yes, the denoiser works for non-Clifford local noise channels.