Non-local differential equations are notoriously difficult to solve. Cell-growth models for population growth of a cohort structured by size, simultaneously growing and dividing, give rise to a class of non-local eigenvalue problems, whose “principal” eigenvalue is the time-constant for growth/decay. These and other novel non-local problems are described and solved in special cases in this paper

Begg, R E

Wake, Graeme

English

ORA - Oxford University Research Archive

Functional differential equations arising in cell-growth

Oxford University Research Archive

Report Number 09/26Functional differential equations arising in cell-growthbyG.C. Wake and R.E. BeggOxford Centre for Collaborative Applied MathematicsMathematical Institute24 - 29 St Giles’OxfordOX1 3LBEnglandFunctional differential equations arising incell-growthG. C. Wake∗and R. E. Begg†July 31, 2009AbstractNon-local differential equations are notoriously difficult to solve. Cell-growthmodels for population growth of a cohort structured by size, simultaneously grow-ing and dividing, give rise to a class of non-local eigenvalue problems, whose“principal” eigenvalue is the time-constant for growth/decay. These and othernovel non-local problems are described and solved in special cases in this paper.1 IntroductionNonlocal calculus is relatively undeveloped. Thus occurs when cause and effect areseparated in space, size, age, or time and give rise to functional differential equations.These problems frequently require unusual boundary and/or initial conditions to en-sure they are well-posed - which would be expected for problems which come frommodelling situations. Some well-known areEXAMPLE. y′(t) = y(t−T ) for all t ∈ R, T > 0, which unlike the correspondinglocal equation when T = 0 has a countably infinite number of linearly independentsolutionsB ={eλit : i = 1, . . . n, . . .}where the λi ’s satisfy the transcendental equationλ = e−λT ,and so there is only one real solution in B.This delay equation, which is well-known of course, needs to have an initial con-dition y(t) = y0(t), −T<t ≤ 0, to make it well-posed. Even more bizarre behaviouremerges if T<0, when the “future dictates the past”.EXAMPLE. y′(t) = y(αt), for all t ∈ R+, α > 0 also has an interesting solutionset which is markedly different if α ≷ 1. This is a special case of the “pantographequation”. See [1] and [2]. If y (0) = y0 is given, it is well-posed if α < 1. For α > 1,it has quite different behaviour.∗Corresponding author: Institute of Information and Mathematical Sciences, Massey Univer-sity at Albany, Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand. e-mail:g.c.wake@massey.ac.nz, Phone: +64 (0) 9 414 0800 Fax: +64 (0) 9 441 8136†University of Waterloo, Canada. e-mail rbegg@maths.uwaterloo.ca.12 The cell-growth modelIt was well-known to biologists prior to the 1980’s that cell-populations, structured bysize (x which is synonymous here to DNA content) which evolve by simultaneouslygrowing, dividing and dying, grow asymptotically towards a steady-size distribution(SSD) which has constant shape with increasing or decreasing size depending on theparameters. In a series of papers, for example [3] this was obtained mathematically byusing a generalised Fokker-Planck equation.nt = (Dn)xx − (gn)x − µn + α2B (αx) n (αx, t) − B (x) n (x, t) , (1)for the evolving, with time t, of number density of cells n (x, t), where D is thedispersion coefficient, g is the per-capita growth rate, µ is the per-capita death rate andB (x) is the frequency of division of cells of size αx into α cells of size x, with α > 1.Although it does not require α to be an integer from a mathematical point of view,usually α = 2.The equation (1) is usually accompanied by the zero-flux condition[D (x)n (x, t)]x=0− g (0)n (0, t) = 0, t > 0, (2)and decay at infinityn (x, t) , nx (x, t) → 0, as x → ∞. (3)The proof that there are SSD-like solutions is established beyond all doubt by ob-serving that there are solutions of the formn (x, t) = e−λty (x) , x, t > 0where λ, y (x) satisfy(Dy)′′(x)− (gy)′(x)−µy+α2B (αx)−B (x) y (x)+λy = 0, x > 0,′ =ddx(4)withy (x) ≥ 0, (Dy)′(0) − (gy) (0) = 0,y (∞) = y′ (∞) = 0}(5)where y (∞) denoteslimx→∞y (x)The first equation in (5) is that of a zero-flux condition.Without loss of generality we can take y (x) to be a probability density function,and so2∫∞0y (x) dx = 1 (6)The problem in equations (4), (5), (6) are what we shall call a non-local, singu-lar Sturm-Liouville eigenvalue problem. The question of the existence of SSD-likesolutions becomes that of : “Is there an eigenvalue λ, and eigenfunction y of this prob-lem?” Further, is it unique within a multiplicative constant (equation (6) eliminates themultiplicity anyway).3 Results3.1 Special casesB (x) = b (a constant), with splitting at any size x. There is only one eigenvalue.Note that there will be other eigenfunctions but not with a mono-signed eigenfunction,which must satisfy∫∞0y (x) dx = 0. (7)The eigenvalue is λ = µ − b (α − 1) with solution for y (x) a Dirichlet series ofthe general formy (x) =∞∑n=0bne−kxn, (8)where k, bn are constants.This function is just one peak and does not exhibit the two peaks formed in practicewith data sets. This means thatµ ≷ b (α − 1) ensures adyinggrowingcohort of cells.It is interesting that this is independent of the growth rate g and dispersion rate D.Which outcome you want depends if the cells are ones you want (muscle cells etc.) orones you do not (tumour cells).3.2 GeneralB (x) Without the requirement of having a mono-signed eigenfunction it is possibleto prove there is a countable number of eigenvalues by restricting the domain to x ∈(0, L) with L arbitrarily large. To get two bumps in the one mono-signed eigenvalue werequire something like B (x) = bδ (x − ℓ), where δ (x − ℓ) is the Dirac-delta functionand ℓ is the cell-splitting size.We expect this result to apply as L → ∞ and conjecture this is so. See [4].33.3 More general non-local eigenvalue problemsThe above can be extended to problems like−y′′ (x) + ay′ (x) + by (x) = λy (g (x)) , 0 < x < ∞y (0) = 0, y (∞) = 0,(9)and g (x) is a continuous function mapping [0,∞) to itself. Letting z = 1 −e−x,Y (z) = y (x), we can then transform the problem (9) into a problem on a finitedomain [0, 1], use the Green’s function for the compact domain to obtain a compactoperator and so the problem can be re-stated as a functional integral equation usingthe Green’s function of the operator on the left-hand side of equation (9). This againshows that there is a countable number of eigenvalues, and each eigenvalue has finitemultiplicity.3.4 StabilityIn most of the above cases we are able to prove that the monosigned SSD behaviouris globally attracting. That is, n (x, t) ∼ Ce−λty (x) for large time irrespective of thesign of λ, and the difference decays faster as t increases.4 DiscussionRecently we came across the non-local heat equation - not based on a physically realproblem which appears well-posed, but is actually not finite for arbitrarily small time:Uxx (1 − x, t) = Ut (n, t) , 0 < x < 1, t > 0U (0, t) = U (1, t) = 0,U (x, 0) given.(10)Then U (x, t) is unbounded as t increases. This shows like all the above problems,that there are astounding unexpected results coming from this and that the earlier non-local problems which appear deceptively simple.ACKNOWLEDGEMENT. The publication was completed while an author (GCW)was a Visiting Fellow at OCCAM (Oxford Centre for Collaborative Applied Mathemat-ics) under support provided by Award No. KUK-C1-013-04, made by King AbdullahUniversity of Science and Technology (KAUST).References[1] J. R. Ockendon, and A. B. Taylor, The dynamics of a current collection systemfor a locomotive. Proceedings of the Royal Society London A(322), pp. 447–468,(1971).[2] L. Fox, D. F. Mayers, J. R. Ockendon, and A. B. Taylor, On a functional differen-tial equation, J.Inst.Math.Appl. 8, pp. 271–307, (1971).4[3] G. C. Wake, S. Cooper, H-Y. Kim, and B. van Brunt, Functional differential equa-tions for cell-growth models with dispersion, Comm.Appl.Anal., 4, pp. 561–573,(2000).[4] B. Basse, G. C. Wake, D. J. N. Wall, and B. van Brunt, On a cell-growth modelfor plankton, Math.Med. & Bio: A Journal of the IMA, 21, pp. 49–61, (2004).5RECENT REPORTS200902/09 Diffusion and permeation in binary solutions Peppin03/09 On the modelling of biological patterns with mechanochemicalmodels: insights from analysis and computationMoreoGaffneyGarcia-AznarDoblare04/09 Stability analysis of reaction-diffusion systems with timem-dependent coefficients on growing domainsMadzvamuseGaffneyMaini05/09 Onsager reciprocity in premelting solids PeppinSpannuthWettlaufer06/09 Inherent noise can facilitate coherence in collective swarm motion Yates et al.07/09 Solving the Coupled System Improves Computational Efficiencyof the Bidomain EquationsSouthernPlankVigmondWhiteley08/09 Model reduction using a posteriori analysis Whiteley09/09 Equilibrium Order Parameters of Liquid Crystals in the Laudau-De Gennes TheoryMajumdar10/09 Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyondMajumdarZarnescu11/09 A Comparison of Numerical Methods used for Finite ElementModelling of Soft Tissue DeformationPathmanathanGavaghanWhiteley12/09 From Individual to Collective Behaviour of Unicellular Organisms:Recent Results and Open ProblemsXueOthmerErban13/09 Stochastic modelling of reaction-diffusion processes: algorithmsfor bimolecular reactionsErbanChapman14/09 Chaste: a test-driven approach to software development for phys-iological modellingPitt-Francis et al.15/09 Block triangular preconditioners for PDE constrained optimiza-tionReesStoll16/09 From microscopic to macroscopic descriptions of cell migration ongrowing domainsBakerYatesErban17/09 The Influence of Gene Expression Time Delays on Gierer-Meinhardt Pattern Formation SystemsSeirin LeeGaffneyMonk18/09 Analysis of a stochastic chemical system close to a sniper bifurca-tion of its mean field modelErban et al.19/09 On the existence and the applications of modified equations forstochastic differential equationsZygalakis20/09 Pebble bed: reflector treatment and pressure velocity coupling Charpin et al.21/09 A finite difference method for free boundary problems Fornberg22/09 Tangent unit-vector fields: nonabelian homotopy invariants andthe Dirichlet energyMajumdarRobbinsZyskin23/09 Morphological instability of a nonequilibrium icecolloid interface PeppinMajumdarWettlaufer24/09 The effect of polar lipids on tear film dynamics AydemirBrewardWitelski25/09 Preconditioning for active set and projected gradient methods assemi-smooth Newton methods for PDE-constrained optimizationwith control constraintsStollWathenCopies of these, and any other OCCAM reports can be obtainedfrom:Oxford Centre for Collaborative Applied MathematicsMathematical Institute24 - 29 St Giles’OxfordOX1 3LBEnglandwww.maths.ox.ac.uk/occam

Mathematical Institute Eprints Archive

Report Number 09/26
Functional differential equations arising in cell-growth
by
G.C. Wake and R.E. Begg
Oxford Centre for Collaborative Applied Mathematics
Mathematical Institute
24 - 29 St Giles’
Oxford
OX1 3LB
England
Functional differential equations arising in
cell-growth
G. C. Wake∗and R. E. Begg†
July 31, 2009
Abstract
Non-local differential equations are notoriously difficult to solve. Cell-growth
models for population growth of a cohort structured by size, simultaneously grow-
ing and dividing, give rise to a class of non-local eigenvalue problems, whose
“principal” eigenvalue is the time-constant for growth/decay. These and other
novel non-local problems are described and solved in special cases in this paper.
1 Introduction
Nonlocal calculus is relatively undeveloped. Thus occurs when cause and effect are
separated in space, size, age, or time and give rise to functional differential equations.
These problems frequently require unusual boundary and/or initial conditions to en-
sure they are well-posed - which would be expected for problems which come from
modelling situations. Some well-known are
EXAMPLE. y′(t) = y(t−T ) for all t ∈ R, T > 0, which unlike the corresponding
local equation when T = 0 has a countably infinite number of linearly independent
solutions
B =
{
eλit : i = 1, . . . n, . . .
}
where the λi ’s satisfy the transcendental equation
λ = e−λT ,
and so there is only one real solution in B.
This delay equation, which is well-known of course, needs to have an initial con-
dition y(t) = y0(t), −T<t ≤ 0, to make it well-posed. Even more bizarre behaviour
emerges if T<0, when the “future dictates the past”.
EXAMPLE. y′(t) = y(αt), for all t ∈ R+, α > 0 also has an interesting solution
set which is markedly different if α ≷ 1. This is a special case of the “pantograph
equation”. See [1] and [2]. If y (0) = y0 is given, it is well-posed if α < 1. For α > 1,
it has quite different behaviour.
∗Corresponding author: Institute of Information and Mathematical Sciences, Massey Univer-
sity at Albany, Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand. e-mail:
g.c.wake@massey.ac.nz, Phone: +64 (0) 9 414 0800 Fax: +64 (0) 9 441 8136
†University of Waterloo, Canada. e-mail rbegg@maths.uwaterloo.ca.
1
2 The cell-growth model
It was well-known to biologists prior to the 1980’s that cell-populations, structured by
size (x which is synonymous here to DNA content) which evolve by simultaneously
growing, dividing and dying, grow asymptotically towards a steady-size distribution
(SSD) which has constant shape with increasing or decreasing size depending on the
parameters. In a series of papers, for example [3] this was obtained mathematically by
using a generalised Fokker-Planck equation.
nt = (Dn)xx − (gn)x − µn+ α
2B (αx)n (αx, t)−B (x)n (x, t) , (1)
for the evolving, with time t, of number density of cells n (x, t), where D is the
dispersion coefficient, g is the per-capita growth rate, µ is the per-capita death rate and
B (x) is the frequency of division of cells of size αx into α cells of size x, with α > 1.
Although it does not require α to be an integer from a mathematical point of view,
usually α = 2.
The equation (1) is usually accompanied by the zero-flux condition
[D (x)n (x, t)]
x=0
− g (0)n (0, t) = 0, t > 0, (2)
and decay at infinity
n (x, t) , nx (x, t)→ 0, as x→∞. (3)
The proof that there are SSD-like solutions is established beyond all doubt by ob-
serving that there are solutions of the form
n (x, t) = e−λty (x) , x, t > 0
where λ, y (x) satisfy
(Dy)
′′
(x)− (gy)
′
(x)−µy+α2B (αx)−B (x) y (x)+λy = 0, x > 0,′=
d
dx
(4)
with
y (x) ≥ 0, (Dy)
′
(0)− (gy) (0) = 0,
y (∞) = y′ (∞) = 0
}
(5)
where y (∞) denotes
lim
x→∞
y (x)
The first equation in (5) is that of a zero-flux condition.
Without loss of generality we can take y (x) to be a probability density function,
and so
2
∫
∞
0
y (x) dx = 1 (6)
The problem in equations (4), (5), (6) are what we shall call a non-local, singu-
lar Sturm-Liouville eigenvalue problem. The question of the existence of SSD-like
solutions becomes that of : “Is there an eigenvalue λ, and eigenfunction y of this prob-
lem?” Further, is it unique within a multiplicative constant (equation (6) eliminates the
multiplicity anyway).
3 Results
3.1 Special cases
B (x) = b (a constant), with splitting at any size x. There is only one eigenvalue.
Note that there will be other eigenfunctions but not with a mono-signed eigenfunction,
which must satisfy
∫
∞
0
y (x) dx = 0. (7)
The eigenvalue is λ = µ − b (α− 1) with solution for y (x) a Dirichlet series of
the general form
y (x) =
∞∑
n=0
bne
−kx
n
, (8)
where k, bn are constants.
This function is just one peak and does not exhibit the two peaks formed in practice
with data sets. This means that
µ ≷ b (α− 1) ensures a
dying
growing
cohort of cells.
It is interesting that this is independent of the growth rate g and dispersion rate D.
Which outcome you want depends if the cells are ones you want (muscle cells etc.) or
ones you do not (tumour cells).
3.2 General
B (x) Without the requirement of having a mono-signed eigenfunction it is possible
to prove there is a countable number of eigenvalues by restricting the domain to x ∈
(0, L)withL arbitrarily large. To get two bumps in the one mono-signed eigenvalue we
require something like B (x) = bδ (x− ℓ), where δ (x− ℓ) is the Dirac-delta function
and ℓ is the cell-splitting size.
We expect this result to apply as L→∞ and conjecture this is so. See [4].
3
3.3 More general non-local eigenvalue problems
The above can be extended to problems like
−y′′ (x) + ay′ (x) + by (x) = λy (g (x)) , 0 < x <∞
y (0) = 0, y (∞) = 0,

 (9)
and g (x) is a continuous function mapping [0,∞) to itself. Letting z = 1 −
e−x,Y (z) = y (x), we can then transform the problem (9) into a problem on a finite
domain [0, 1], use the Green’s function for the compact domain to obtain a compact
operator and so the problem can be re-stated as a functional integral equation using
the Green’s function of the operator on the left-hand side of equation (9). This again
shows that there is a countable number of eigenvalues, and each eigenvalue has finite
multiplicity.
3.4 Stability
In most of the above cases we are able to prove that the monosigned SSD behaviour
is globally attracting. That is, n (x, t) ∼ Ce−λty (x) for large time irrespective of the
sign of λ, and the difference decays faster as t increases.
4 Discussion
Recently we came across the non-local heat equation - not based on a physically real
problem which appears well-posed, but is actually not finite for arbitrarily small time:
Uxx (1− x, t) = Ut (n, t) , 0 < x < 1, t > 0
U (0, t) = U (1, t) = 0,
U (x, 0) given.

 (10)
Then U (x, t) is unbounded as t increases. This shows like all the above problems,
that there are astounding unexpected results coming from this and that the earlier non-
local problems which appear deceptively simple.
ACKNOWLEDGEMENT. The publication was completed while an author (GCW)
was a Visiting Fellow at OCCAM (Oxford Centre for Collaborative AppliedMathemat-
ics) under support provided by Award No. KUK-C1-013-04, made by King Abdullah
University of Science and Technology (KAUST).
References
[1] J. R. Ockendon, and A. B. Taylor, The dynamics of a current collection system
for a locomotive. Proceedings of the Royal Society London A(322), pp. 447–468,
(1971).
[2] L. Fox, D. F. Mayers, J. R. Ockendon, and A. B. Taylor, On a functional differen-
tial equation, J.Inst.Math.Appl. 8, pp. 271–307, (1971).
4
[3] G. C. Wake, S. Cooper, H-Y. Kim, and B. van Brunt, Functional differential equa-
tions for cell-growth models with dispersion, Comm.Appl.Anal., 4, pp. 561–573,
(2000).
[4] B. Basse, G. C. Wake, D. J. N. Wall, and B. van Brunt, On a cell-growth model
for plankton, Math.Med. & Bio: A Journal of the IMA, 21, pp. 49–61, (2004).
5
RECENT REPORTS
2009
02/09 Diffusion and permeation in binary solutions Peppin
03/09 On the modelling of biological patterns with mechanochemical
models: insights from analysis and computation
Moreo
Gaffney
Garcia-Aznar
Doblare
04/09 Stability analysis of reaction-diffusion systems with timem-
dependent coefficients on growing domains
Madzvamuse
Gaffney
Maini
05/09 Onsager reciprocity in premelting solids Peppin
Spannuth
Wettlaufer
06/09 Inherent noise can facilitate coherence in collective swarm motion Yates et al.
07/09 Solving the Coupled System Improves Computational Efficiency
of the Bidomain Equations
Southern
Plank
Vigmond
Whiteley
08/09 Model reduction using a posteriori analysis Whiteley
09/09 Equilibrium Order Parameters of Liquid Crystals in the Laudau-
De Gennes Theory
Majumdar
10/09 Landau-De Gennes theory of nematic liquid crystals: the Oseen-
Frank limit and beyond
Majumdar
Zarnescu
11/09 A Comparison of Numerical Methods used for Finite Element
Modelling of Soft Tissue Deformation
Pathmanathan
Gavaghan
Whiteley
12/09 From Individual to Collective Behaviour of Unicellular Organisms:
Recent Results and Open Problems
Xue
Othmer
Erban
13/09 Stochastic modelling of reaction-diffusion processes: algorithms
for bimolecular reactions
Erban
Chapman
14/09 Chaste: a test-driven approach to software development for phys-
iological modelling
Pitt-Francis et al.
15/09 Block triangular preconditioners for PDE constrained optimiza-
tion
Rees
Stoll
16/09 From microscopic to macroscopic descriptions of cell migration on
growing domains
Baker
Yates
Erban
17/09 The Influence of Gene Expression Time Delays on Gierer-
Meinhardt Pattern Formation Systems
Seirin Lee
Gaffney
Monk
18/09 Analysis of a stochastic chemical system close to a sniper bifurca-
tion of its mean field model
Erban et al.
19/09 On the existence and the applications of modified equations for
stochastic differential equations
Zygalakis
20/09 Pebble bed: reflector treatment and pressure velocity coupling Charpin et al.
21/09 A finite difference method for free boundary problems Fornberg
22/09 Tangent unit-vector fields: nonabelian homotopy invariants and
the Dirichlet energy
Majumdar
Robbins
Zyskin
23/09 Morphological instability of a nonequilibrium icecolloid interface Peppin
Majumdar
Wettlaufer
24/09 The effect of polar lipids on tear film dynamics Aydemir
Breward
Witelski
25/09 Preconditioning for active set and projected gradient methods as
semi-smooth Newton methods for PDE-constrained optimization
with control constraints
Stoll
Wathen
Copies of these, and any other OCCAM reports can be obtained
from:
Oxford Centre for Collaborative Applied Mathematics
Mathematical Institute
24 - 29 St Giles’
Oxford
OX1 3LB
England
www.maths.ox.ac.uk/occam


Functional differential equations arising in cell-growth

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