Population Fluctuations of Growing Clones in Age-Dependent Stochastic Models:Supplemental Material

Evgeny B. Stukalin1,2, Ivie Aifuwa2,3, Jin Seob Kim2, Denis Wirtz2,3, and Sean X. Sun1,2

1Department of Mechanical Engineering and 2Physical Sciences in Oncology

Center, The Johns Hopkins University, Baltimore, Maryland 212183Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218

A: VARIANCE AND GROWTH RATE FOR CONSTANT CELL DIVISION AND CELL DEATH RATES.

Consider the birth-death process with constant transition probabilities per unit time for cell division kb and celldeath kd and kb > kd. The explicit consideration of the age of cells a is not needed to derive the average growth ratev and variance (or dispersion) σ2. Let us define P (n, t) as the probability to find n cells at time t. The governingmaster equation is

dP (n, t)

dt= kb(n− 1)P (n− 1, t) + kd(n+ 1)P (n+ 1, t)

− (kb + kd)nP (n, t) (S1)

where n = 0, 1, 2, . . . with with P (−1, t) ≡ 0. The average size of the population is N(t) =∑∞n=0 nP (n, t). From Eq.

(S1) it follows

dN(t)

dt=

∞∑n=0

kbn(n− 1)P (n− 1, t) + kd n(n+ 1)P (n+ 1, t)

−∞∑n=0

(kb + kd)n2P (n, t)

=

∞∑n=0

kb(n+ 1)nP (n, t) + kd(n− 1)nP (n, t)

−∞∑n=0

(kb + kd)n2P (n, t) (S2)

The terms proportional to n2 cancel out and we arrive

dN(t)

dt=

∞∑n=0

(kb − kd)nP (n, t) = (kb − kd)N(t) (S3)

The solution of Eq. (S3) is N(t) = N0e(kb−kd)t where N0 is the initial number of cells. Let us compute the dispersion

defined as σ2(t) =∑∞n=0 n

2P (n, t) − N2(t) = 〈n2〉 − N2. Again multiplying Eq. (S1) by n2 and summing, the firstterm in the variance satisfies the following equation

d〈n2〉dt

=

∞∑n=0

kbn2(n− 1)P (n− 1, t) + kd n

2(n+ 1)P (n+ 1, t)

−∞∑n=0

(kb + kd)n3P (n, t)

=

∞∑n=0

kb(n+ 1)2nP (n, t) + kd(n− 1)2nP (n, t)

−∞∑n=0

(kb + kd)n3P (n, t) (S4)

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The terms proportional to n3 cancel out and we obtain

d〈n2〉dt

=

∞∑n=0

2(kb − kd)n2P (n, t) +

∞∑n=0

(kb + kd)nP (n, t)

= 2(kb − kd)〈n2〉+ (kb + kd)N(t) (S5)

For the second term of the variance, we have

dN2

dt= 2N(t)

dN(t)

dt= 2(kb − kd)N2(t). (S6)

Finally, we combine the first and the second terms by Eq. (S5) and Eq. (S6)

dσ2(t)

dt≡ d

dt

(〈n2〉 −N2(t)

)= 2(kb − kd)σ2(t) + (kb + kd)N(t) (S7)

The final Eq. (S7) can be solved easily by assuming some initial condition, e.g. σ2(0) = 0, since all growth trajectoriesalways come from N0 cells at time t = 0, so P (n, 0) = δn,N0 . The solution is

σ2(t) =kb + kdkb − kd

N0

[e2(kb−kd)t − e(kb−kd)t

]≈ kb + kd

kb − kdN0 e

2(kb−kd)t for long times (S8)

The relative population fluctuation is defined as r = σ(t)/N(t) , which at long times converges to constant valueobtained from Eq. (S8):

r =σ(t)

N(t)=

√1

N0

kb + kdkb − kd

(S9)

If there is no cell death kd = 0 the relative standard deviation is r = 1/√N0.

For this model with constant kb and kd, P (n, t) at long times is roughly a Gaussian function of n with width σ(t)given in Eq. (S8). With the age-dependent models (Fig. S9 and S10), the cell number distribution is also roughlyGaussian. However, the width is no longer predicted by Eq. (S8).

B: SOLUTION OF THE VON FOERSTER EQUATION FOR INITIALLY SYNCHRONIZEDPOPULATION

The solution of the von Foerster equation (Eqs. (2-5) in the main text) with age-dependent cell birth k(a) andconstant cell death kd is given by

N(t) =

∫ ∞0

n(a, t) =

∞∑j=1

Nj (S10)

N1(t) = N0 exp

[−∫ t

0

k(a) da

]exp (−kd t) (S11)

Nj(t) = 2

∫ t

0

Nj−1(t− a)ω(a) exp (−kd a) da (S12)

where N(t) is total number of cells at time t and j = 2, 3, 4, . . .. The normalized probability density function ofgeneration times ω(a), which is known before hand (

∫∞0ω(a)da = 1), is related to the rate of cell division through

Eq. (6) of the main text.

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C: SIZE FLUCTUATIONS FOR COMPETITIVE CELL POPULATION UNDER GROWTH

Consider the birth-death process with constant probabilities per unit time for cell division kb, cell death kd, andadditional cell death γ = γ0(n − 1) where γ0 is constant and n is number of cells. In this case age of cells is notimportant for our analysis. The governing master equation is

dP (n, t)

dt= kb(n− 1)P (n− 1, t) + kd(n+ 1)P (n+ 1, t)

+ γ0n(n+ 1)P (n+ 1, t)

− (kb + kd + γ0(n− 1))nP (n, t) (S13)

where n = 0, 1, 2, . . . with with P (−1, t) ≡ 0.Let us assume that initial state of the system is defined as P (n, 0) = δn,N0 where N0 is initial number of cells at

time t = 0, which implies that σ(0) = 0. Further analysis shows that eventually, the population approaches steadystate with Nss = (kb−kd)/γ0. Assume that steady state population is the one of macroscopic size Nss 1. Considertwo cases of initial conditions: (A) |N0 − Nss| ' Nss (e.g., N0 = 1 as in our simulations) and (B) N0 is of order ofNss, i.e., |N0 −Nss| Nss.

Consider case B first when |N0 −Nss| Nss. In this case one can use Ω-expansion of governing master equation.[1, 2]

n = Ωφ(t) +√

Ωx, (S14)

where φ(t) is non-fluctuating variable, x is fluctuating continuous variable, and Ω = 1/γ0 is system size parameter.Using this transformation of variables one may find that φ(t) satisfies ordinary differential rate equation

dφ

dt= (kb − kd)φ− φ2 (S15)

The solution is

φ(t) =φ(0)e(kb−kd)t

1 + φ(0)(kb−kd)

[e(kb−kd)t − 1

] (S16)

The probability distribution function Π(x, t) =√

ΩP (Ωφ(t) +√

Ωx) satisfies the Fokker-Planck equation

∂Π(x, t)

∂t= [kd − kb + 2φ(t)]

∂

∂xxΠ +

1

2[kd + kb + φ(t)]φ

∂2

∂x2Π (S17)

We can find the expressions for the mean x(t) and for the variance x2(t) without solving Eq. (S17). By integratingby parts one can find that these two functions obey the following equations

dx

dt= [kb − kd − 2φ(t)]x (S18)

and

dx2

dt= 2 [kb − kd − 2φ(t)]x2 + φ(t) [kb + kd + φ(t)] (S19)

At steady state φ(t) = φss = kb − kd, so

dx

dt= (kb − kd) x (S20)

and

dx2

dt= −2 (kb − kd)x2 + 2kb (kb − kd) (S21)

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with initial conditions x(0) = x0 = N0√γ0 and x2(0) = x20 = N2

0 γ0. The solutions are

x(t) = x0e−(kb−kd)t (S22)

and

σx(t) ≡√x2(t)− x(t)

2=√kb(1− e−2(kb−kd)t) (S23)

In the long time limit, the mean size of population Nss = φssΩ = (kb−kd)/γ0 1. The time dependence of standarddeviation is described by σ(t) = σx(t)

√Ω = σx(t)/

√γ0 with σx(t) in Eq. (S23). Thus, if the initial size of population

N0 is of order Nss the standard deviation of the average population size grows monotonically from zero to its steadystate value σss =

√kb/γ0. If kd = 0 then σss =

√Nss. The mean population size Nss and its variance σss at steady

state calculated by Ω-expansion correspond to stationary Gaussian distribution. [1, 2]However, if |N0 −Nss| ' Nss corresponding to our simulations (N0 = 1) one may not use Ω-expansion to describe

full time dependence of σ(t) since this method is applicable if the system is not far from steady state. Basically,we can analyze two regimes: (I) At early times N(t) ' N0 death events due to cell “competition” are very rare. Inthis case σ(t) ∼ e(kb−kd)t grows exponentially as for population without competitive cell death. (II) In long timelimit N(t) = Nss. In this regime σ(t) equals to its steady state value σ(t) = σss. This regime has been consideredabove for the case B. Between these two regimes of exponential growth of σ(t) and steady state regime we have quitecomplicated crossover regime which is not analytically tractable.

Simulations with constant rates kb, kd, and γ0 (kd = 0 is taken for simplicity as for simulation in age-dependentcase) show that σ(t) is not monotonic function of time and goes through the maximum if N0 Nss, e.g., N0 = 1,as in our simulations. The numerical values of parameters are kb = 0.03516 (a.u.)−1, kd = 0, and γ0 = 5 × 10−6

(a.u.)−1. The parameter γ0 is the same as for simulations of age-dependent competitive population while kb is chosensuch that it would give the same steady state size of population Nss as for age-dependent growth. The data for 120growth trajectories are analyzed at steady state regime for time interval from t = 440 a.u. to t = 474 a.u. (the resultsare not shown). We find that Nss = 7016 with standard deviation σss = 94.9. The size distribution fits normaldistribution quite well. These values can be compared to theoretically expected ones Nss = (kb − kd)/γ0 = 7032and σss =

√kb/γ0 = 83.9. Also note that unlike σss in the steady state regime the maximum value of dispersion

σmax = 1592 for constant cell division rate is essentially (a factor of 4) higher than σmax = 407 for “typical” (non-exponential) generation time distribution (see Fig. 7D). The difference in values of σss also exists, however it is muchsmaller (a factor of 1.4). For the case of constant rates we used the same simulation algorithm as for age-dependentcell division.

D: AGE-DEPENDENT SIMULATION ALGORITHM

The initial age distribution consists of N0 cells with ages (a1, . . . , aN0) at time t = 0. The time increment ∆t is

a small constant (∆t = 0.02 a.u. in the most of simulations). The index j defines each cell in the list. The statedescriptor qj describes the two possible states of the cell: if the j-th cell is alive but undivided qj = 1 otherwise qj = 0.At each time step i at time t = i∆t, we go through the list of cells from j = 1 to j = Ntotal where Ntotal is the totalnumber of cells. Ntotal = Nalive +Nlost +Ndivided, which counts alive cells Nalive, lost cells Nlost and cells that haveterminated after their division Ndivided. The list of cells Nj is formed from the previous time step i− 1 and is usedfor the i-th time step in unmodified form. The list of cells Sj that will be used for the next step i + 1 is formedfrom Nj by updates during step i (cell divisions, cell deaths, and age tracking). For each alive cell (qj = 1) tworandom numbers are generated 0 < rs < 1 (s = 1, 2). If r1 < f1(aj) = exp−(k(aj) + kd)∆t, the j-th cell remainsalive but undivided, otherwise j-th cell dies, or divides. If no events happen at time step i, the cell age is incrementedby ∆t. However, if r1 > f1(aj), the second random number is used to choose between mitosis and cell death. Celldivision occurs if r2 < f2(aj) = k(aj)/(k(aj) + kd), provided r1 > f1(aj). In this case two newborn cells are addedto the end of the new list Sj. After each division, the updated Stotal = Stotal + 2 while the number of alive cellsincreases by one Nalive = Nalive + 1. The states of the newborn cells are described as aStotal−1 = aStotal

= 0 andqStotal−1 = qStotal

= 1 whereas the state descriptor of the mother cell becomes qj = 0. If r2 > f2(aj) and r1 > f1(aj),the j-th cell dies and its state descriptor changes from one to zero qj = 0. The number of dead cells increases by oneNlost = Nlost + 1 and the number of alive cells decreases by one cell Nalive = Nalive − 1. The transient size of Sjlist in this algorithm is Stotal = 2(Nalive +Nlost)−N0 where N0 is the initial number of alive cells at t = 0.

When condition j = Ntotal is satisfied, the i-th time step is completed. The number of alive and lost cells are givendirectly by Nalive and Nlost. Alternatively, the number of alive cells can be computed as Nalive =

∑Stotal

j=1 qj . The

5

simulation is continued until t = tmax, or Nalive = Nmax when tmax and Nmax are the desired time, or populationsize at the end of the run, respectively.

Typically, the population is allowed to grow from a single newborn cell (a = 0), or initial age of a starting cell ischosen randomly from uniform distribution from 0 to τ . The average number of cells N(t) is obtained by averagingover large number of growth trajectories (“runs”). In the absence of cell death the number of runs was 400 fordistribution I, and 1000 for other distributions II-IV. In these cases the average number of cells at the end of eachgrowth trajectory was about 20000. For the constant non-zero cell death kd > 0 the number of growth trajectories canbe smaller, but not less than 200. For competitive populations the data from 120 runs are analyzed. The standard

deviation σ(t) of the mean population size is computed as σ(t) =√∑R

k=1(Nk(t)−N(t))2/(R− 1), where R is number

of growth trajectories, Nk(t) is number of cells at time t for k-th growth trajectory, and N(t) is average number of

cells N(t) =∑Rk=1Nk(t)/R. The average growth rates v are obtained from the fits of N(t) to exponential functions.

The steady state age distributions are obtained by analyzing 25 growth trajectories with final ages recorded at theend of each run. For details see caption to Fig. 3. The bin size δa = 0.5 a.u. is used to build up the normalized agedistribution function g(a).

E: CELL DIVISION TIME MEASUREMENT FOR HUMAN DERMAL FIBROBLAST

Cells: Human Dermal Fibroblast cells (GM00038) were obtained from Coriell Cell Repositories. GM00038 cellswere cultured in Dulbecco’s modified Eagle’s medium supplemented with 10% (v/v) fetal bovine serum (FBS, Hyclone)and 0.1% Penicillin/Streptomycin (Sigma).

Division Time Distribution: 1x104 GM00038 cell were seeded (sparsely) in a 35mm glass bottom dish (Mattek)and allowed to spread for 24 hours in an incubator at 37C and 5% CO2. Cells then were imaged using a NikonTE2000E Microscope at low magnification (×10) for 48hours and maintained at 37C and 5% CO2. Cell divisionswere obtained by tracking the duration of a single cell from one division until the division of both daughters. > 130cells were assessed.

[1] van Kampen, N.G. 1981 Stochastic processes in physics and chemistry. Amsterdam, The Netherlands, Elsevier Science Ltd.[2] Bergersen, B. 2003 Stochastic processes in physics. Lectures on the web. Available: http://www.phas.ubc.ca/∼birger/510n4/

and . . . /510n5/.

SUPPLEMENTAL FIGURES

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Fig. S 1: Growth curves (color lines) for age-dependent cell division rate k(a) and constant cell death rates kd. Each curverepresents the average over 20 runs for synchronized colonies (N0 = 10 cells of age a = 0). The growth rates v are obtained fromexponential fits (thin black lines). The inset depicts the average growth rate v as a function of cell death rate kd (symbols).The solid line is the linear fit v = v0− bdkd with bd = 0.989± 0.005 and v0 = 0.0532± 0.0001. The generation time distributionis the shifted gamma distribution with parameters α = 6, β = 0.5 a.u., and a0 = 10 a.u. (τ = 13 a.u.).

Fig. S 2: The average number of cells as functions of time for five different generation time distributions in the absence of celldeath (see legends). For distributions I-IV, the cells are grown from a single cell of zero age a = 0. For exponential distributionV, for which the age of a starting cell does not matter, we use the Gillespie algorithm. The average generation time for alldistributions I-V is the same (τ = 20 a.u.) but the coefficients of variation of τ for I-V are different. The generation timedistributions are described by shifted gamma distributions with parameters: α = 6 (I-IV), and β = 0.91, 1.82, 2.86, 3.33 a.u.and a0 = 14.55, 9.09, 2.86, 0 a.u. for I, II, III, IV, respectively. The inset shows the growth curves at early times. The growthcurves are the average of 400 runs for distribution I and of 1000 runs for distributions II-V.

7

Fig. S 3: The average number of cells as a function of time in the absence of cell death for synchronized and asynchronouscell populations for generation time distributions I (400 runs) and II (1000 runs) (see legends and also caption to Fig. S2).In all cases the populations are grown from a single cell. For synchronized populations, the initial age of the cell is zero. Forasynchronous colonies, the age of a starting cell is chosen randomly from 0 to τ for each run. The inset depicts the samegrowth curves at early times. The effect of the initial age distribution on the growth rates is negligible (at long times) but thepre-factors of exponential fits are different.

Fig. S 4: Average number of cells N(t) grown from a single cell of age a = 0 for generation time distribution I (see captionto Fig. S2) from 400 runs (thick blue line). Magenta lines represent the functions equal to N(t) plus and minus the standarddeviation σ(t). The red line is the analytical solution of the von Foerster equation for synchronized populations. The thincurves with maximums represent the contributions Nj(t) to N(t) from different generations: j = 1,2, etc. (see Appendix B).

8

Fig. S 5: The average growth curves in the absence of cell death kd = 0 and for constant cell death rate kd > 0. The generationtime distribution is II (see caption to Fig. S2). The growth curve with kd = 0 is taken from Fig. S2. The growth curveswith kd > 0 are averaged from 200 runs. The cell populations are grown from a single cell with zero age a = 0 (in all cases).The ratios of the number of lost cells Nlost to the number of alive cells Nalive are shown as functions of time for kd > 0. Thethin black lines represent theoretical dependencies obtained from integration of dNlost/dt = kdNalive(t) with initial conditionNlost(0) = 0, where Nalive(t) is the long time exponential ansatz.

Fig. S 6: The effect of initial number of cells N0 the relative standard deviation of the mean for synchronized cell colonies (seelegends). The generation time distribution is II (see caption to Fig. S2). The initial ages of all N0 cells are taken to be zeroa = 0. The cell death rate is kd = 0.005 (a.u.)−1. The inset shows the long time relative standard deviations (σ/N)lim vs N0

(symbols). The solid line is the power law fit (σ/N)lim = bN0N−X0 with exponent X = 0.52± 0.01 and bN0 = 0.56± 0.03.

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Fig. S 7: The average growth curves for synchronized cell populations (200 runs for each curve). The generation timedistribution is II (see caption to Fig. S2). The cell colonies are grown from different number of cells N0 (see legends) in thepresence of the constant cell death kd = 0.005 (a.u.)−1. The ratios of the number of lost to the number of alive cells are shownas functions of time for each case (the color of the curve in the inset corresponds to the one in the main figure). The thin blackline represents the theoretical dependence obtained as explained in the caption to Fig. S5.

Fig. S 8: The effect of initial number of cells N0 on the relative standard deviation for asynchronous cell colonies in absenceof cell death (see legends). Generation time distribution II is used (see caption to Fig. S2). The initial ages of N0 cells aretaken randomly from uniform distribution from 0 to τ for each run. The curve for N0 = 4 is obtained for 400 runs. The curvefor N0 = 1 is taken from Fig. 6 (1000 runs) and re-scaled by a factor 1/2 (thin black line) according to expected power lawdependence σ(t)/N(t) ∼ 1/

√N0.

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Fig. S 9: Size distributions of competitive populations at steady state. 120 growth trajectories are analyzed for the timeinterval from t = 400 a.u. to t = 474 a.u. for different values of γ0 = 2× 10−5, 1× 10−5, and 5× 10−6 (a.u.)−1 (see Fig. 7A).The mean sizes of the population at steady states are Nss = 1759, 3519, and 7024 cells with the standard deviations σss =29.4, 45.1, and 68.3, respectively. “Theoretical” values obtained as the ratio v/γ0 are Nss = 1762, 3525, 7050. The solid linerepresents the Gaussian distributions with the same parameters.

Fig. S 10: Size distribution of growing cell populations at different times (see legends) for 1000 growth trajectories in theabsence of cell death (symbols). The data correspond to synchronized cell populations. These distributions can be describedby “moving” Gaussian distributions. (A) The population size distributions at different times for generation time distributionII. (B) The same distributions for wider generation time distribution III (see caption to Fig. S2). The correspondence betweenthe symbols and the legends in the panel B are the same as in panel A.

Fig. S 11: Distributions of times tn to reach n 1 cells from a single cell of zero age a = 0 in the absence of cell death arepresented as histograms for different n (see legends). These distributions can be constructed from the weighting of populationsize distributions at fixed times from Fig. 10S. The data are analyzed for 1000 growth curves from simulations: (A) Timedistributions to reach n = 200, 1000 and 5000 cells for generation time distribution II. (B) The same dependencies for widergeneration time distribution III. See caption to Fig. S2 for parameters of distributions II and III in panels A and B.