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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_2796_Библиотеки_им_академика_М_И_Перельмана
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0.5
0.4
0.3
0.2
of time
fraction
0.1
0
0 8 15
0.3
0.2
0.1
0
0 8 15
0.1
0
0 8 15
XXX
Skewness
Weeks
FIGURE 7.8 Damage distributions are skewed to the right and skewness drops with age.
XX
young old
time (weeks) time (weeks)
0.6
0.4
0.2
(Wk,Wk+8)
correlations
-0.2
816
24 32 40 48 56 64 72
Weeks
FIGURE 7.9 Damage uctuates rapidly in young and becomes more persistent with age as seen in
rising autocorrelation time.
THE SATURATING REMOVAL MODEL CAN
EXPLAIN SENESCENT CELL DYNAMICS
ese dynamical features of senescent cells can be explained by our simple truck model
for aging, which we call the saturating removal model, developed by Omer Karin in his
PhD with me (Karin et al. 2019). Omer scanned a wide class of models and found that the
essential features that a model needs to explain the senescent-cell dynamics are precisely
the truck model. In fact, these dynamical features are enough to constrain the possible
models into a single minimal candidate.
e rst feature is two timescales, one fast and one slow: X is produced and removed on
a timescale that is much faster than the rise in DPU which occurs on the timescale of the
lifespan. is separation of timescales allows us to write an equation for the rate of change
of X in which the production and removal rates vary slowly and depend on age. e model
also includes stochastic noise. us,
Aging and Saturated Repair ◾ 167
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dX
e production rate of X rises linearly with age in the model:
=−production removaln+ oise
dt
where we use τ for age and t for time to make sure that we understand that there are two
timescales: a fast scale (days-weeks) in which damage reaches steady state, and a slow timescale (years) over which production rate
e removal of X is carried out by removal processes, such as NK cells that kill senescent
cells. If this removal worked at a constant rate
thus be −
quasi-steady-state solution is a linear rise of X with age, X
data which shows an accelerating rise with age (Figure 7.5).
with the number of senescent cells. In other words, removal saturates; senescent cells
inhibit their own removal. Such a drop could be due to several processes: immune cells
that remove senescent cells are downregulated if they kill too oen, and they can be inhibited by factors that the senescent cells produce. e drop of removal rate can also be simply
due to a saturation eect, in which the removing cells become increasingly outnumbered
by senescent cells.
bition due to secreted factors and for saturation (see Exercise 7.4)
β
X. However, this does not match the data. e equation is =−
Instead, the model explains the accelerating rise in X by a removal rate that drops
To model the saturation, we use a Michaelis–Menten form which is good both for inhi-
production =
ητ
Removal =
ητ
changes.
β
per senescent cell, the removal term would
dX
ητ β
=ητ/β. is is ruled out by the
st
X
β
κ
+ X
dt
X, whose
where
β
is the maximal senescent-cell removal capacity, in units of senescent cells/time, and
κ
is the concentration of X at which they inhibit half of their own removal rate. e removal
rate per senescent cell thus drops with senescent cells abundance, (Figure 7.10), and
the number of senescent cells lost per unit time is that rate times X, namely .
Combining production and removal, we obtain a model for the rate of change of X:
(7.1) =−
Note that this model assumes that maximal removal capacity
age. Adding such a decline, namely
simplicity, we ignore this possibility and recall that NK cell numbers do not drop strongly
with age in humans.
β
κ
+ X
β
κ
dX X
ητ β
dt
β
τ
, generally leaves the conclusions the same. For
()
κ
+ X
β
does not decline with
X
+ X
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removal rate
+X
X
FIGURE 7.10 Removal rate per senescent cell drops with senescent-cell abundance X.
Let’s compute the steady state of X. On the fast timescale of weeks, the production rate
ητ
can be considered as constant. Setting dX/0dt = in Eq. 7.1, we nd that the (quasi-)
steady-state X of is
κητ
(7.2) X
≈
st
βη−τ
us, X
closer to zero. e rise of X accelerates and diverges at a critical age
rises linearly with age at rst. en, the term on the bottom becomes closer and
st
τ
=
βη
/ (Figure 7.11).
c
Figure 7.4 shows a good agreement between data in dots and the model in black.
When X levels rise high enough, they reach levels not compatible with life. us, the
τ
=
βη
critical age
/ where X diverges is a rough approximation for the mean lifespan.
c
is equation indicates that lifespan can be extended by increasing the repair capacity
η
or by reducing the senescent-cell production rate
. More trucks or slower construction of
houses extend longevity.
X
st
5
4
3
2
1
C
β
,
FIGURE 7.11 Mean senescent cells X accelerates and diverges at a critical age.
Aging and Saturated Repair ◾ 169
κ
ξ
dx
dt
x
x
st
Potential U(x)
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ADDING NOISE TO THE MODEL EXPLAINS THE VARIATION
BETWEEN INDIVIDUALS IN SENESCENT-CELL LEVELS
If this model was all there was, then all individuals would age at the same rate and die at
the same age. In other words, so far, the model does not explain why genetically identical
organisms dier in the number of senescent cells. We need to describe the uctuations of
X over time which produce the dierences between individuals. To understand these stochastic features of the dynamics, we need to consider the noise in the model.
e simplest way to model noise is to add a white-noise term
adds and removes small amounts of X randomly, with mean zero and a variance described
by the parameter
. e factor 2 is for convenience in the equations below. e noise
describes uctuations in production and removal due to internal or external reasons such
as injury, infection, and stress (cortisol). Variations in sleep, which aects repair, can also
play a role. In fact, we don’t currently know what the noise describes. White noise is a
convenient way to wrap up our ignorance in a mathematical object that we can work with.
We arrive at the main model of this chapter and the next, a stochastic dierential equa-
tion called the saturating removal model:
ητ β
(7.3)
=−++2
. is white-noise term
Let’s take a nice deep sigh of relief to celebrate making it to this point.
THE SATURATING REMOVAL MODEL CAPTURES
THE VARIATION BETWEEN INDIVIDUALS
We will now use this model to understand the dynamics of senescent cells, and then to
understand the origin of the Gompertz law. Let’s begin with the variation in X between
individuals at a given age. To do so, we need to compute the damage distribution, P(X).
ere is an elegant way to compute it. Historically, this approach comes from analysis
of particles undergoing Brownian motion under the inuence of forces, known in physics as Langevin equations. I like this approach because it has a graphic interpretation in
which X is the position of a particle rolling down a bowl-shaped contour (Figure 7.12).
noise
XX
FIGURE 7.12 Damage dynamics can be likened to a particle in a bowl-shaped potential well jiggled
by noise.
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To describe the bowl, one recasts Eq. 7.3 using a potential function U(X). e potential
function provides the shape of the bowl. e slope of the potential is the restoring force
that pushes the particle back to the equilibrium at the bottom of the bowl: the slope −dU/
dU X
dX is dened as production minus removal, =−
ητ β
dX X +
. At the bottom of the bowl
κ
the slope is zero, and thus production equals removal, so that the bottom is the position of
the steady state.
=−
e required potential is UX
()
βητβ
()
ing the derivative –/dU dX to verify that it gives
e particle rolls down to the bottom of the bowl and noise randomizes its position
XX−+
κκ
log
ητβ
− .
()
X +
. You can check this by tak-
X
κ
(Figure 7.12). e distribution of X that we seek is then analogous to the Boltzmann law
that applies to particles in a potential under thermal uctuations. e noise amplitude
−U /
~e
plays the role of temperature, and we obtain
Px
()
. e mathematical details are
provided in Solved Exercise 7.1 at the end of the chapter.
At young ages, the senescent cells are like a particle in a steep potential well because
removal is not saturated and pushes hard to reduce senescent cells. e particle is jiggled by
noise in the narrow well, and so P(X) is narrowly distributed around a low mean damage
(Figure 7.13). With age, the potential well opens to the right, as production rate rises. e
mean damage increases and so does its variance (Figure 7.13).
e saturating removal model thus predicts a distribution of damage that we can nd by
−U /
~e
plugging in the potential into the Boltzmann term
PX
()
to obtain
(7.4) PX
()()
βκ
κ
+ Xe
− tX
βη
−
()
is distribution rises with X, reaches a peak, and then falls exponentially. It is skewed to
the right and matches the skewed distributions observed in the mouse data (Figure 7.8,
upper panel, blue lines).
is distribution provides an improved estimate for the average X that takes noise into
account,
〈〉X ≈+
κητβ
(7.5)
U(X)
FIGURE 7.13 With age the bowl widens to the right, increasing damage mean and variance.
X
()
time
Young
−
ητ
()
X
time
Old
X
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e mean X rises with age and diverges at age
with age and diverges at
(7.6)
τ
:
c
σ
≈
β
τ
= . e standard deviation of X also rises
c
η
2
κβ
+
βη−τ
is rise in the mean and standard deviation matches the observed rise with age of the
standard-deviation in mice (Figures 7.5 and 7.6, blue lines).
Notably, the model even captures the fact that variation rises more slowly than the mean –
the phenomenon of reducing relative heterogeneity. e ratio between the standard deviation
and the mean of X, the coecient of variation CV, drops with age as observed
2
+
CV =≈ ~ .
σ
X
κβ
κηττ
1
+
is reducing heterogeneity is a distinguishing feature of the model. Most simple stochastic models show a constant CV.
e model also captures the increasing correlation times with age. At young ages, the bowl
X
is steep. us, if X is away from steady-state
, it returns to it quickly (Figure 7.13). At old
st
age, in contrast, the bowl is almost completely at. e trajectory of the particle is dominated
by noise, with little restoring force coming from the steepness of the bowl (Figure 7.13). Hence
X
individuals that stray away from
have a weak restoring force and stay away for longer times.
st
Such increasing correlation times have a general name in physics, “critical slowing
down.”. ey are a mark of an approaching phase transition. In the canonical example of
a phase transition, the boiling of water, large and slow uctuations in density can be seen
near the boiling point. In other areas of science, slowing down of uctuations can be a
warning sign of a big transition. Examples include climate uctuations before an ice age,
or ecological uctuations before a species extinction (Dai 2015). In our case, the phase
transition is to death.
e mouse data allows estimating the four model parameters,
para meters a re approximately
η
==410
−−42
days~0.15/year/day,0
η, β,κ
βκ
, and ε. e best t
.3/day, ==1,0.1,
in units where the average senescent-cells level in young mice is 1. e rough estimate of lifespan
β
τ
= ~2years is about right for mice.
c
η
ese parameters give a specic prediction for the half-life of a senescent cell. e
half-life is about 5 days in young mice and rises to about a month in old mice (25 days in
22-month old mice) because the trucks get overloaded. ese predictions were intriguing
enough to test experimentally, which Omer Karin did with senescent-cell expert Valery
Krizhanovsky (Karin 2019). In young mice, the senescent-cell half-life was
old mice (22-month-old), removal was much slower, with an estimated half-life of about a
month. ese measurements agree well with the predictions of the model.
5±1 days. In
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GOMPERTZ MORTALITY WITH SLOWDOWN IS FOUND IN THE MODEL
We can now nally ask why do genetically identical organisms die at dierent times? What
explains the exponential increase in risk of death with age (the Gompertz law), and the
deceleration at very old ages (Figure 7.14)?
To connect senescent-cell dynamics to mortality, we need to know the relationship
between senescent-cell abundance and the risk of death. e precise relationship is currently unknown. Let’s therefore simply model death when senescent-cell abundance
X
exceeds a threshold level
system or a tipping point such as sepsis. us, death is a rst-passage time process, when
senescent cells cross X
We use this threshold-crossing assumption to illustrate a way of thinking, because
it provides analytically solvable results. Other dependencies between risk of death and
senescent-cell abundance, such as Hill functions with various degrees of steepness, provide
similar conclusions.
(Figure 7.15). e threshold represents a collapse of an organ
C
.
C
1
10
10
-1
-2
Gompertz
law
h ~ e
deceleration
)
-1
-3
10
-4
Hazard, h (years
10
020406080 100
Age, (years)
FIGURE 7.14 e risk of death rises exponentially according to the Gompertz law and decelerates
at very old ages.
X
time of
death
x
C
age
FIGURE 7.15 Death is modeled to occur when X crosses a threshold X
.
c
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Again, there is an elegant way to solve this rst passage time problem and obtain the risk
of death as a function of age. To estimate the probability that X crosses the death-threshold
X
, we apply an approach which is analogous to the rate of a chemical reaction crossing an
c
energy barrier ∆G, namely the Boltzmann factor, e
ε plays the role of temperature k
potential U at X
and at the steady-state value Xst, ∆=UU
c
ability per unit time for X crossing X
T, and the energy barrier is the dierence between the
b
, namely the risk of death that we call the hazard, is
c
−∆U /
h ≈e
∆G
−
kT
b
. In our case, the noise amplitude
XU
−
()
Xt. us, the prob-
()
cs
is equation is called Kramer’s equation in the eld of stochastic processes. An intuitive explanation is that the particle in the well needs to climb a potential dierence of
∆=UU
“kicks” provided by the noise of size
Since you need
goes as
XU
−
()
−∆U /
e
Xtto fall o into the death region (Figure 7.16). It needs to climb using
()
cs
∆U/
(Figure 7.17).
. Each noise kick can be either to the right or le.
kicks, all in the right direction, the chance is exponentially small and
U
U
X
st
FIGURE 7.16 e particle needs to climb a potential dierence of ∆U with noise kicks of size ε in
X
C
order to fall o into the death region.
U
death
U
“steps” ~ e
X
st
FIGURE 7.17 Death requires ∆U/ε kicks in the same direction, an event with exponentially small
X
C
probability.
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Now because the production rate of senescent cells rises linearly with age, it becomes
easier and easier to reach the threshold X
. As the potential bowl opens up, the “energy bar-
c
rier” drops linearly with age, and risk therefore rises exponentially with age. e outcome
for the risk of death (see Solved Exercise 7.1) is up to age-independent factors:
κη
+X
τ
()
(7.7)
h
τβ
≈−
()
()
ητ
κβ
+1
e
C
is is a big moment. e hazard rises exponentially with age as e
ατ
with an exponent α,
called the Gompertz aging rate, given by
+ X
κη
()
α
=
C
.
We derived the Gompertz aging rate in terms of molecular parameters: noise, damage
production rate and death thresholds. e aging rate is proportional to the slope of damage
η
production with age,
rises with the death threshold X
. It is inversely proportional to the noise amplitude . Aging rate also
(see Solved Exercise 7.1 for further discussion).
c
e Gompertz law is thus intimately related to the linear rise with age of the production
ητ
rate. If the production rate rose with age not linearly
2
ητ
, we would not get the Gompertz law but instead a hazard that rises exponentially with
age squared, e
2
ατ
.
is solution also explains the observed deceleration of the hazard rate at very old ages
(when
ητβ
≈ ). is deceleration is due to the prefactor
τ
=
βη
as age approaches the critical age
/ . Fundamentally, slowdown occurs because near
c
but as age squared, for example,
κβ
+1
βη−τ
()
which becomes small
the critical age production and removal rates are nearly equal and cancel out, and noise
becomes the primary driver of X. erefore, damage dynamics resemble a random walk,
with a nearly constant probability per unit time of crossing the threshold.
1
e saturating removal model thus analytically reproduces the Gompertz law, including
the deceleration at old ages (Figure 7.18). It gives a good t to the observed mouse mortality
curve using parameters that agree with the experimental half-life measurements and longi-
X
tudinal senescent-cells data. e inferred threshold for death is
threshold X
is about 17 times larger than the mean senescent-cell level in young individuals.
c
=±17 2, meaning that the
c
In brief, genetically identical mice die at dierent times due to noise amplied by accumulating damage producing units and a saturating damage removal.
THE HUMAN GOMPERTZ LAW IS CAPTURED AS WELL
So far, we calibrated the model on mouse data. Let’s now use it to study human mortality
curves. We can ask which of the model parameters changes between mice and humans to
provide the dierence in their lifespans. A good description of human hazard curves, corrected for extrinsic mortality, is provided by the same parameters as in mice, except for a
60-fold slower increase in
shows the model with added extrinsic mortality at a level indicated by the dashed line, in
comparison to the observed mortality.
η
,the slope of senescent-cell production with age. Figure 7.19
Aging and Saturated Repair ◾ 175
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mouse mortality
)
-1
0.010
0.005
0.001
5.x10
hazard rate (weeks
experiment
model
-4
100 125 150
age (weeks)
FIGURE 7.18 e saturating removal model analytically captures the Gompertz law and decelera-
tion of mortality in mice.
human mortality
0.100
experiment
model
)
-1
FIGURE 7.19 e saturating removal model with additive constant extrinsic mortality reproduces
the Gompertz law and deceleration of mortality in humans.
e smaller value of η in humans can be due to improved DNA maintenance and
enhanced damage repair in humans compared to mice. us, there are fewer DPUs at a
given age, and each DPU gives rise to fewer senescent cells. Fewer houses each making less
garbage.
0.010
0.001
-4
10
hazard rate (years
20 40 60 80 100
h et( - t)
age (years)
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