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Calculus from the past and at a distance: Applications in cancer cell growth

Primers in Math/Stat, UoC 26 th April

Graeme Wake1 *

with

Ali Ashher Zaidi1, Bruce van-Brunt2. and Helen Byrne3,

1 Centre for Mathematics-in-Industry, Institute of Natural and Mathematical Sciences, Massey University Auckland2. Institute of Fundamental Sciences, Massey University Manawatu.3. OCCAM, University of Oxford

*With Funding from

*

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This is an expository talk aimed at non-experts with certainly some new Maths and an important application

underpinning decision support!

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Outline

• Delay equations• Biology background• A new generic model• Key mathematical challenges Watch this space!!!

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1. Time delay: Arise in population models. First example; “R18 Calculus”. y‘(t) = y(t-T), t ≥ 0; y(t) = yo(t), -T < t ≤ 0.

Questions: 1. Well-posed?2. What space is y in?3. Solution + ?? PRIZE!!!!!

Questions: 1. Well-posed?2. What space is y in?3. Solution + ?? PRIZE!!!!! Under 30s only.

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1. Time delay: Arise in poulation models. First example; “R18 Calculus”. x‘(t) = x(t-T), t ≥ 0; x(t) = xo(t), -T < t ≤ 0.

Questions: 1. Well-posed?2. What space is y in?3. Solution + ?? PRIZE!!!!!

Questions: 1. Well-posed?2. What space is x in?3. Solution + ?? PRIZE!!!!! Under 30s only.

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Questions: 1. Well-posed? Yes2.What space is x in? Depends on xo(t).3.Solution + ?? x(t) = ∑∞ cn exp(λnt), where the λn are in the

n=0 set {λ : λ = exp(- λ T) } ~ N ko ≈ 0.56 (when T = 1 ); Re(λn ) < 0, n >0 and occur in complex conjugate pairs: messy calculation.

:

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Transcendental equation λ = exp(- λ), when T = 1 ;

λ1 = Re(λ), λ2 = Im(λ) satisfy λ1 = exp(- λ1 ) cos(λ2 ) (i)

and λ2 = - exp(- λ1 ) sin(λ2 ). (ii) Only one real solution.

Infinite number of complex conjugate solution pairs.

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Solutions of real part (red (i)) and imaginary (blue (ii)) part of exp(-λ) – λ= 0

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2. Multiple delays ( very recent)• There can be multiple delays in systems.• Cell-cycle population dynamics, as in Simms, Bean, & Koerber (in BMB).• Abstracted by me to dx/dt = A x(t-[T]), ………………………….(*)where x(t) is a column vector of functions{ x1(t), ....xi(t),......xn(t) }

A is a nxn (constant matrix)T = (Ti j ) and [T] = Ti j , j = 1,….n when x → xi .

(Explains [T] ??), that is dxi /dt = ∑ ai j xj (t- Ti j )

j

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The Human Cell Cycle

M

G1

S

G2

11 hours(highly variable)

10 hours

4 hours

1 hour

Senescence

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Cell- Cycles

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Cell-cycle dynamicsfrom Simms (PhD thesis UWA 2012)

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Core model (as in Simms et al)Letting NGa (t ),NGb (t ), . . . ,NMb (t) represent the number of cells in the Ga,Gb,. . . , Mb phases at time t respectively, and recalling that dx(t)/dt

represents the rate ofchange of x(t) with respect to time, we have equations as

γ

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Note equation 1c. One component has 2 components with different delays,

Easily accommodated.

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[In the Simms et al model T = (Ti j), the elements are independent of i.]How to solve completely:dx/dt = A x(t-[T]), …………….(*)where x(t) is a column vector of functions?? Quite a challenge?

Prize???Not in the textbooks!!!!

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Solve: dx/dt = A x(t-[T]), … Aha…. try x(t) = eλt cThen, λ c = B(λ) c , where B(λ) = (ai j exp( -λTi j ) ), and A = (aij ) Wow…..

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So λ ε {λ : det(B(λ) – λI) = 0 } = ΛThis is a “nonlinear in λ” eigenvalue problem with a countable infinity of “eigenvalues”, but c(λ) is the corresponding “eigenvector” of B(λ).

So the general solution of dx/dt = A x(t-[T]), is therefore

(expected to be, proofs coming) x(t) = ∑ exp(λt ) c(λ)

λ ε Λ

Wake GC & Byrne HM, ANZIAM J to appear June 2013

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2. Biology background: Structured by size

• The ability of cells to divide asymmetrically is essential for generating diverse cell types during development.

• The past 10 years have seen tremendous progress in our understanding of this important biological process.

• The relevance of asymmetric cell division for stem cell biology has added a new dimension to the field, and exciting connections between asymmetric cell division and tumour genesis have begun to emerge.

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• Cell divisions producing two daughter cells that adopt distinct fates are defined as asymmetric.

• In all organisms, ranging from bacteria to mammals, in which development has been studied extensively, asymmetric cell divisions generate cell diversity.

• Asymmetric cell divisions can be achieved by either intrinsic or extrinsic mechanisms, Intrinsic mechanisms involve the preferential segregation of cell fate (read “size”) determinants to one of two daughter cells during mitosis.

• Asymmetrically segregated factors that bind cell fate determinants and orient the mitotic spindle may also be necessary to ensure the faithful segregation of determinants into only one daughter cell.

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Intrinsic and extrinsic mechanisms in asymmetric cell divisions.

Hawkins N , Garriga G Genes Dev. 1998;12:3625-3638

©1998 by Cold Spring Harbor Laboratory Press

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Intrinsic and extrinsic mechanisms in asymmetric cell divisions.

Intrinsic factors indicated in red are distributed asymmetrically in the progenitor cell and inherited by one daughter cell (Z). These factors can be cell fate determinants that specify the fate of the daughter cell that inherits them, factors that distribute cell-fate determinants, or factors that orient the spindle. Extrinsic mechanisms require cell signalling that can occur (1) between daughter cells, (2) between a daughter cell and surrounding cells, or (3) between the precursor cell and surrounding cells.

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Taxol effect: 6 weeks “in vivo”

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Our ObjectiveSSD behaviour: n( . ,t) evolves like:

SSD Behaviour

0123456789

10

0 1 2 3 4Cell Size

Cel

l Cou

nt

t2

t1

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0 0.5 1 1.5 2 2.5 3 3.50

100

200

300

400

500

600

700

800

x (Relative DNA Content)

Cel

l Cou

nt

NZM13 DNA profile 0 hours after the addition of Taxol

ModelData

Doubling Time:

Model: 72 hours

Data: 76 hours

Melanoma Cells in Vitro

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3. Notation and Our Model• Independent variables t time in hours : [t] = T x “size” (=fate) in ngrams: [x] = M Dependent variable n(x,t) number density of cells w.r.t x [n] = M-1

M(t) = = # cells in a ≤ x ≤ b. Size x

Transfer function W(x,ξ) = # cells of size x from division of one cell of size ξ : [W] = M-1 .

0 τ x ξ

| |||

b

a

dxtxn ),(

b

a

dxtxn ),(

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More notationTransfer function W(x,ξ) = # cells of size x from

division of one cell of size ξ : [W] = M-1

Frequency at which cells divide into two or more daughter cells b(x) : [ b ] = T-1

Per capita death rate µ(x) : [µ] = T-1

Per capita growth rate g(x) : [g] = M T-1

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The equation.Analogy

• Think of n as a “concentration”, as a function of “position” x;• g as a velocity;• Do a balance on n…….The ultimate non-local p.d.e 0 τ x ξ

•B.c, i. c: n(0,t) = n(∞,t) =0; n(x,0) = no(x) (arbitrary)

)(),())()2(),()((

))1((),(),()()(

0

debitstxndeathsdx

xWxb

creditsdtnxWbgnxt

n

x

x

(2) (1)

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The splitting function W(x,ξ)

• What sort of W(x,ξ)’s are feasible???• Answer….If a cell of size x splits to give many cells of size τ, 0 < τ < x, conservation of mass demands (*)

This is an ill-posed partial Volterra equation.

x

dxWx0

),(

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A simplification to The EquationUsing (*) we get The Equation simplified as

This satisfies “mass” conservation, and has to be solved with the b.c and i.c. n(0,t) = n(∞,t) = 0, n(x,0) = no(x)

and with admissible W(x,ξ) satisfying (*).

),())((),(),()()(

txnxbdtnxWbx

gn

t

n

x

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Examples of W(x,ξ) to date…1. W(x,ξ) = 2 δ(ξ/2 – x) : δ = the Dirac-delta

function, the symmetrical special case: two daughter cells of equal size.

2. W(x,ξ) = W(x – ξ) = δ(x – ξ) , not real in the sense that the cells are not really dividing, but it satisfies (*)!!!!

3. W(x,ξ) = U(x)/V(ξ), the separable case: (noting U,V have different dimensions)

then (*) gives:

x

dUx

xV0

)(1

)(

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4. The most realistic case binary/asymmetric: α > β >1 constants І І І І І

0 z =x/α z = x/β [ x ] ξ = βx :ξ = αx out inThusW(x, ξ) = δ(ξ/α – x) + δ(ξ/β – x) The mass conservation equation

implies = 1, or β =

x

dxWx0

),(

1

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3. Key mathematical challenges

• Solve THE equation for various W;• Look for separable solutions n(x,t) = e -λt y(x)• Get a non-local first-order singular eigenvalue

problem, 1. With W(x, ξ) = 2 δ(ξ/2 – x), g, b constant (gy)’(x) = 4b y(2x) – (b + μ)y(x) + λ y(x); y(0) = y(∞) = 0 ; y ≠ 0.

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4. The binary/asymmetric case:With W(x, ξ) = δ(ξ/α – x) + δ(ξ/β – x),

g, b constant and 1/α + 1/β = 1 we get (gy)’(x) = b α y(α x) + b β y(βx) –

-(b + μ)y(x) + λ y(x); y(0) = y(∞) = 0 ; y ≠ 0.

This is a two term “pantograph equation”!!!

This is an eigenvalue problem.What are the

eigenvaues/eigenfunctions??

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Conjectures, for some cases, yet to be fully understood.

Spectrum = {λn : n ε N} ;

Eigenfunctions {yn(x) : n ε N } “complete” in

some sense; n(x,t) = Σ cn exp(−λnt) yn(x) n ε N Note: there maybe some non-discrete/continuous spectra, as in y’’(x) + λy(x) = 0, 0 < x <∞; y(0) = 0, │ y(∞ )│ < ∞. Then SSD- behaviour is n(x,t) ~ exp(−λmt) ym(x),

where λm = the eigenvalue with smallest real part.

If Re (λm ) > 0, cohort decreasing;

Re (λm ) < 0, cohort increasing:

Both asymptotically of constant shape. This is seen experimentally.

Future Extensions

• Let growth be stochastic, using the stochastic growth given by dx = gdt + σdX,(A Langevin equation, where X is a Wiener

process, with variance σ2 ) and using the Fokker-Planck equivalence, interpreting n(x,t) as a p.d.f, we add to the LHS

where D = σ2/2, making The Equation parabolic, not hyperbolic. The same challenges will arise.

.

2

2 )(

x

Dn

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Questions/Comments

Acknowledgments:• Professor Helen Byrne (Oxford) for pointing out the need for conservation on the splitting function W, thus giving the condition (*)• Support from

dxWxx

),(0