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Exercise 7.3: Intuitive derivation of the Boltzmann-like form of the steady-state
distribution
Consider a stochastic process of the form =+vx
velocity of x. Dene the potential U(x) by =−vx
state, the probability distribution is Px
Solution: Consider a large number of particles moving along a one-dimensional pipe.
ey diuse with diusion coecient
. e particle density at steady state is P(x). e ux at point x due to the velocity eld
vx
()
is the velocity times the density:
law of diusion, which shows a diusive ux from high to low densities proportional to
−
the gradient:
to zero: vx
us, at steady state, in regions where velocity is large, the density P(x) shows a steep opposing slope so that diusion ux – the eects of noise – can balance velocity ux.
Exercise 7.4: Removal of senescent cells based on saturating their own removal
process
Senescent cells are removed by immune cells such as NK cells, which we will denote by R.
ere are a total of
appreciably with age. e R cells meet senescent cells, denoted X, at rate k
plex [R X] which can either fall apart at rate k
us, RX+→
dP /dx. At steady state, total ux is zero, so that the two uxes must sum
Pd−=
()
Pd/0x . us, = . e solution is Px
R
removing cells in the body, and that this number does not change
T
RX R.
[]
vx
()
dX
()
and are also swept along the pipe by a velocity eld
Px
()
vx
dP
dx
()
dt
dU
dx
=−Pe0xp
. e ux due to diusion can be found by Fick’s
Px
()
()
or end up killing senescent cells at rate v.
off
2
ξ
. e function vx
. Explain intuitively why, at steady
()
Ux
()
.
()
is called the
()
=−Pe0xp
to from a com-
on
Ux
()
a. Explain the following dynamic equation for the complex:
b. Use the fact that R cells can be either free or in a complex, so that RRXR
show that the removal rate of senescent cells is
c. Compute the maximal removal capacity β, and the half-way saturation point k.
Explain intuitively.
Exercise 7.5: No repair
Consider an accumulation process of damage with constant production and no removal
dR
X
[]
=−kRXvkRX
dt
Removal =
+
off
X
[]
+=, to
[]
T
on
()
β
kX+
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dX
ηξ
=+
2
dt
a. What is the mean damage X as a function of age?
b. What is the distribution P(X)?
c. What is the hazard assuming that death occurs when X>X
? Is there a Gompertz law?
c
Exercise 7.6: Age-dependent reduction in repair capacity
Consider a process in which damage is produced at a constant rate
η
, and removal does not
saturate. Removal rate per cell drops with age,
dX
ηββτ
dt
=+
()
X 2
−+.
1
ξ
a. What is the mean damage X?
b. What is the distribution P(X) at age?
c. What is the ratio of mean and standard deviation of X:
<X>/
?
σ
d. What is the hazard, if death occurs when X Xc> ? Is there a Gompertz law?
Exercise 7.7: Deterministic model
Assume that the Gompertz law arises not from stochastic eects, but instead from individual dierences, set at birth, in X production and removal parameters, in which each indi-
vidual i has its own noise-free equation =−
crosses threshold X
. What distribution of production and removal parameters
c
dX
dt
ηβ
X. Death is modeled to occur when X
ii
η
,β. can
ii
provide the Gompertz law? What patterns of aging does this model not explain?
Exercise 7.8: Parameter eects
What is the eect on the hazard curve of the saturating removal model of a change in each
β,ηκ
of the parameters
,, ? Plot examples of hazard curves to demonstrate your answer.
Exercise 7.9: Senescent-cell half-life
Show that in the saturating removal model, the half-life of a senescent cell is
t
=+log2
1/2
κβ
()
()
/
ββ
−
()
τ
Exercise 7.10: Critical slowing down
Read (Scheer et al. 2009).
a. How does critical slowing down relate to the model?
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b. Suggest a phenomenon beyond those discussed in Scheer which might show critical
slowing down and suggest an experiment or measurement to test this.
Exercise 7.11: (Challenging question) General model
η
Damage is produced at rate
X,τ and removed at rate
()
β
X,τ. e equation is
()
a. What is the steady-state distribution at age τ?
b. What is the risk of death as a function of age, modeled by rst passage time of a
threshold X
c. Under which conditions does risk of death go as the Gompertz law?
Exercise 7.12: Strehler and Mildvan (1960) model for the Gompertz law
Strehler and Mildvan (1960) (SM) proposed a phenomenological process for the Gompertz
law. Organisms are assumed to start with an initial survival capacity, termed the vitality
V, that declines linearly with age
vitality loss per unit time. Animals experience random external challenges or insults with
a mean frequency K. Challenges have random magnitudes, exponentially distributed with
an average magnitude D that expresses the average deleteriousness of the environment.
Death occurs when the magnitude of a challenge exceeds the remaining vitality. A review
of the SM theory can be found in (Finkelstein 2012).
a. Show that these assumptions produce the Gompertz law
and b.
?
c
dX
ητ
XX,,
=++
()
dt
τ
as V(τ) = V0(1 − Bτ), where B indicates the fraction of
βτ
()
2
ξ
τ
h ae
()
b
=τ. Calculate a
b. What similarities and dierences does this theory have with the saturated removal
model?
Exercise 7.13: Heterochronic parabiosis
Parabiosis is the surgical joining of two mice so that they share circulation. When joining
a young and an old mouse, known as heterochronic parabiosis, the young mouse shows
signs of aging whereas the old mouse rejuvenates. Senescent cell abundance decreases in
the old mouse, whereas it increases in the young mouse (Karin and Alon 2021).
a. Explain these eects using the model. Hint: the trucks from the young mouse can
help the old mouse.
b. e survival curve of the old mouse is made steeper and that of the young mouse less
steep compared to a control experiment joining two equal-aged mice. Explain using
the model.
c. Develop an extended model that describes heterochronic parabiosis. Assume that
senescent cells stay in the tissues and are not shared by the mice, but that the immune
Aging and Saturated Repair ◾ 189
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cells that remove them are shared through the circulation. Plot senescent cells in each
of the two mice as a function of time aer joining.
Exercise 7.14: Aging rates of dierent organs
Based on the model, would you expect that dierent organs in the same individual will age
at similar rates or dierent rates? Explain (100 words).
Exercise 7.15: Numerical solution and decreasing twilight
a. Write a computer program to solve the SR model.
b. Suppose that death occurs when X rst crosses Xc, and illness occurs when X rst
crosses X
, with Xd<Xc. In 100 runs of the simulation, record the time of illness and
d
time of death.
c. Does average remaining lifespan after illness, known as twilight, increase with
age of illness onset or decrease? Answer: twilight should decrease with the age
of illness.
d. Explain the decreasing twilight phenomenon in the model (50 words).
is exercise requires numerically simulating a stochastic ordinary dierential equation.
e main idea is to use Euler’s method (Chapter 1, Exercise 1.6), with timestep
dt, and to
add a noise term consisting of a random number times a timestep dt . e square root is
because noise acts like a random walk or diusion process, which moves a mean distance
dt in a time interval dt.
of ~
Here is a simple algorithm to get you started. Suppose you wish to solve the equations
dx
=+fx
()
2
ξ
with initial condition x(0) = 0.
x(0) = 0 (initial condition)
t(0) = 0 (time = 0)
N = 1000 (total time steps)
dt = 0.1 (0.1s intervals)
for i = 1:N
t(i) = t(i−1)+dt
r(i) = r andom_ numbe r
r(i) dt
dx(i) = f(x(i−1)) dt +
2
x(i) = x(i−1)+dx(i)
end
NOTES
1 Note that the analytical approximation begins to be inaccurate when ηt approaches β,
and simulations of the full model are needed to compute the hazard curve at old ages.
Simulations show that the rise of the hazard curve slows and converges to a constant hazard at very old ages.
2 A parameter set for ies is β =1 /hour, ϵ=1 /hour and η = 0.03/hour day, X
life-extending diet have a lower production slope η = 0.02 /hour day.
= 15. Flies on a
c
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FURTHER READING
Karin et al. 2019. “Senescent cell accumulation slows with age providing and explanation for the
Gompertz law.”
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DOI: 10.1201/9781003356929-14
Chapter 8
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Age-Related Diseases
exclusively at old age. ese age-related diseases
M
include cancer, osteoarthritis, failure of specic organs such as heart failure, kidney
failure, and lung failure, and neuro-degenerative diseases such as Alzheimer’s disease and
Parkinson’s disease.
In this chapter, we will understand why age is the major risk factor for these diseases
and explore the universality of their dynamics. We will also discuss treatment. ese diseases are currently treated one by one, and we will discuss how future medicine can take a
major step forward by treating aging itself in order to address all of these diseases at once.
Age-related diseases are diverse and aect dierent systems. It is therefore striking that
they share a common pattern in their incidence curves. Incidence is the probability to get
the disease at a given age. It is calculated by considering 100,000 people without the disease
at age t and asking how many will be diagnosed over the following year.
And now for the pattern shared between hundreds of age-related diseases:
e incidence of age-related diseases rises exponentially with age and drops at very
old ages (Figure 8.1). e slope of exponential increase is similar for dierent diseases, but
not identical, around 3%–8% per year.
Understanding this exponential rise is a major aim of this chapter. We need to understand why age 20 is dierent from age 70 in ways that make these diseases so much more
likely. We will also understand why incidence drops at very old ages.
Another goal of this chapter is to explain the causes of several diseases of unknown
origin. In doing so we will see mathematical analogies between diseases. is will form
columns in the periodic table of diseases featured in the next chapter of our book.
DISEASES CAUSED BY THRESHOLD CROSSING OF SENESCENT
CELLS HAVE AN EXPONENTIAL INCIDENCE CURVE
To understand age-related disease incidence, we will use a simple model based on the
senescent cell theory of the previous chapter. is model was developed by Itay Katzir during his PhD with me (Katzir et al. 2021).
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FIGURE 8.1 e incidence of age-related diseases rises exponentially with age and drops at very
old ages.
e basic idea is that diseases of old age are due to a phase transition in which aging
pushes a parameter of a physiological circuit across a threshold. Once the threshold is
crossed, the circuit behavior changes dramatically: cells grow without control as in cancer
or die without control as in degenerative diseases.
Aging indeed aects the parameters of physiological circuits. In particular, senescent
cell load X induces systemic inammation and reduces regeneration, which changes circuit
parameters. Above a certain level of senescent cells, the circuit undergoes a bifurcation-its
steady state becomes unstable and pathology emerges. erefore, in the model, a disease
occurs when senescent cells cross a threshold that is specic for each disease. We call this
threshold the disease threshold X
Although each disease has its own threshold X
.
d
, the underlying senescent cell dynam-
d
ics are common to all diseases. ese dynamics are described by the saturating removal
model of Chapter 7, Eq. 7.3. When the concentration of senescent cells X crosses the disease
threshold, the individual gets the disease (Figure 8.2). Each individual crosses the threshold at dierent times, due to the stochastic nature of the dynamics of senescent cells.
e time of disease onset is therefore the time when senescent cell concentration rst
crosses the threshold X
– a rst-passage time problem.
d
Conveniently, in the previous chapter we already solved this rst-passage-time problem.
e solution is an exponential hazard curve – the Gompertz law – that slows at very old
ages. e probability of crossing the threshold X
rises exponentially with age, with an
d
exponential slope of approximately
α
/≈ηX
d
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Disease onset
person 1
person 2
person 3
X
d
Senescent cells, X
020406080
Age[years]
FIGURE 8.2 In the disease-threshold model, a disease occurs when senescent cells cross a disease-
specic threshold.
where η and are the senescent cell production and noise parameters.
is explains the exponential rise of disease incidence curves. Since diseases have dier-
X
ent exponential slopes, each disease has its own threshold
not exceed X
=17, otherwise the model would predict that death precedes the disease,
death
. e disease threshold must
d
and we would not observe the disease.
DECLINE OF INCIDENCE AT VERY OLD AGES IS
DUE TO POPULATION HETEROGENEITY
If this were all, everyone would cross the disease threshold in the model and get the disease. In reality only a fraction of people ever do. is is where the second parameter in the
φ
model comes into play – only a fraction
φ
ranges between zero and one. Some conditions are rare with low φ, others like hyper-
tension and osteoarthritis are common, with
susceptibility depends on genetic and environmental factors, as we will discuss.
e susceptible fraction stems from the notion of population heterogeneity in the elds
of epidemiology and genetics. People dier in their risk for a given disease. To model this,
φ
we assume that only a fraction
of the population has a low disease threshold Xd. e
remaining population has higher values of the disease threshold that are not reached during normal aging. We call these the non-susceptible fraction of the population.
e susceptible fraction explains the decline of incidence curves at very old ages. Recall that
incidence is computed from the population without the disease. At very old ages, most of those
that are susceptible have already had the disease. is results in the decline in incidence rate.
e model thus has two parameters for each disease: the disease threshold and the susceptibility. Let’s solve the model for the incidence curve to see where the rise and fall originate (see Solved Exercise 8.1 for more details about the approximations involved). e idea
is that incidence I(t) is approximately equal to the hazard h(t) – the probability to cross the
disease threshold X
at age t, multiplied by the disease-free survival curve F(t) – the frac-
d
tion of the population who still did not get the disease. us It
of the population are susceptible. e parameter
φ
exceeding 0.1. e precise value of the
= ht
.Ft
()
()
Since h rises
()
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Logincidence
0 20 40 60 80
Age
FIGURE 8.3 In the disease threshold model, incidence rises exponentially and drops at very old
ages.
and F declines, their product is a curve with a maximum. Writing disease-free survival in
terms of hazard results in an equation for the incidence
t
−∫ht()dt
It
()
=φht
and by plugging in a Gompertz-like hazard h=he
e
()
0
α
t
0
, we obtain an analytical incidence
()
formula
h
0
()
α
t
(8.1) =
Ih0ee
φ
()
α
t
−−e
1
()
α
At rst, incidence rises exponentially (Figure 8.3), until at very old ages the last term dominates, since it is an exponential of an exponential, and incidence plummets.
φ
Note that susceptibility
simply multiplies the incidence in Eq. 8.1 and thus deter-
mines its overall height; the shape of the incidence curve, including its slope, intercept,
and age of peak incidence, is determined by a single parameter – the disease threshold X
d
Using the saturated removal model of the previous chapter, we can nd how the disease
threshold determines the shape parameters in Eq. 8.1: to a good approximation, the slope
α
=−0.0090Xd.02 and the hazard intercept is logh
is
of disease thresholds X
between 10 and 16 (Katzir et al. 2021).
d
Armed with Eq. 8.1, we can now nd the best-t values of X
04=−.14 Xd for the relevant range
()
10
and φ for a given empirical
d
incidence curve and see how well the disease-threshold model captures the data.
THE MODEL DESCRIBES WELL THE INCIDENCE
CURVES OF AGE-RELATED DISEASES
To test this model requires a global set of incidence curves. We turn to the large medicalrecord database from Clalit health services that we used in Chapter 3 on hormone seasonality. e data includes about 900 disease categories, each found in the records of at
least 10,000 people. e categories are international disease codes, called ICD9 level 2. Of
these, about 200 diseases rise at least 20-fold between ages 30 and 80, and can be dened
as strongly age-related diseases.
.
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