Transcript An Ecological Perspective - Arizona State University

Biological Stoichiometry of
Tumor Dynamics:
An Ecological Perspective
• Yang Kuang
• Department of Mathematics and Statistics
• Joint work with James J. Elser and John Nagy.
• Work is partially supported by NSF grant DMS0077790
Cancer Facts
More than 1.2 million Americans develop cancer
each year. A new cancer is diagnosed every 30
seconds in the United States.
Lung and prostate cancer are the top cancer killers
for men in the United States. Lung and breast
cancer are the top cancer killers for women in the
United States.
One in two men in the U.S. will be diagnosed with
cancer at some time during his lifetime. One in
three women in the US will be diagnosed with
cancer at some time during her lifetime.
War on Cancer
• President Nixon signed the National Cancer
Act into law on December 23, 1971, declaring,
"I hope in the years ahead we will look back on
this action today as the most significant action
taken during my Administration."
• The Question is: "Are we winning the war?"
Malignant Maths
Jan 22nd 2004. The Economist print edition
Mathematical models aid the understanding of cancer
•
•
PRACTITIONERS of so-called hard sciences—those backed up by the
mathematical rigour of formulae and equations—have traditionally looked down on
the squishy end of research. That disdain has evaporated a bit over recent years, as
government money has migrated from physics to biology and medicine. But it is
disappearing as biologists show that they can be just as quantitative as their hardedged colleagues.
And one example is in the field of cancer research. According to Hans Othmer, a
mathematician at the University of Minnesota, in Minneapolis, who has written a
review of the subject forthcoming in the Journal of Mathematical Biology, a rapid
growth in the understanding of the microscopic processes behind cancer is allowing
useful mathematical models of the disease to be developed. Indeed, the field is
booming, which is why the ponderously named Discrete and Continuous
Dynamical Systems—Series B, another scientific journal, is devoting a special
issue to the subject in February.
The first picture of a ribosome. Cate et al. (1999) Science
285: 2095-2104.
Elevated CO2 and Leaf Mites
• Study: Joutei, Roy, Van
Impe and Le brun published
in 2000.
• Elevating the level of CO2
caused larger leaf biomass
with less nitrogen content.
The leaf mites’ progeny
were reduced by 34% and
49% in the 1st and 2nd
generations and later stages
of development were
reduced.
6000
mg C
High (Light/Nutrient), K=6000 mgC, P=100 mgP
5000
4000
y
A
3000
Region II
2000
1000
Region I
0
10
20
30
40
50
days
60
K x
0
Very High (Light/Nutrient), K=12000 mgC, P=100 mgP
y
12000
mg C
10000
8000
B
6000
Region II
4000
Region I
2000
0
10
20
(________) prey
30
40
50
60
70
80
days 90
0
K x
(- - - - - - -) predator
Figure 1. All parameter values are as in Table 1, initial conditions x=600 mgC, y=500 mgC. K=6000 mgC
in A and K=12000 mgC in B. In both numerical runs, the solution is attracted to stable steady state in
Region II. Energy enrichment of the system stabilizes predator-prey interactions in A. Further increase in
K leads to deterministic extinction of the predator. Respective phase planes on the right show that in A the
positive equilibrium in Region II is locally asymptotically stable, since prey nullcline decreases steeper
then the predator’s; in B, prey nullcline lands on the right of the second x-intercept of predator nullcline
making the equilibrium locally asymptotically stable.
Tumor
The Growth Rate Hypothesis
natural selection
on growth rate
cellular investment
(ribosome content)
biochemical investment
(RNA:protein)
Body C:N:P
resource
nutrient
food quality
trophic
competition
recycling
constraints
efficiency
on growth /
reproduction
The first picture of a ribosome. Cate et al. (1999) Science
285: 2095-2104.
Based on: Elser, J.J., D.R. Dobberfuhl, N.A. MacKay, and J.H. Schampel.
Organism size, life history, and N:P stoichiometry: toward a unified view of
cellular and ecosystem processes. BioScience 46: 674-684.
Are P-rich Animals Also RNA-rich Animals?
1:1
From: Dobberfuhl, D.R. 1999. Elemental stoichiometry in crustacean zooplankton: phylogenetic patterns, physiological
mechanisms, and ecological consequences. PhD dissertation, Arizona State University, Tempe, AZ.
Characteristics of nucleic
acids extracted from
normal and tumor tissues
• Int J. Gynecol Cancer 2002 Mar; 12(2): 171-176
• T Y Chu et al.
• As expected, tumor tissues have roughly twice as
much of P Iphosphate) content compare to
normal tissues.
• Normal tissues have about 1% (dry weight) P
and tumor tissues have about 2% P.
Cancer and healthy cells
compete for phosphorous
• Many lines of evidence lead to the conclusion that
ribosomes and therefore phosphrous, is a important
commodity in cancer cells. The bigger and more active
the nucleous, the faster cancer cells proliferate in vivo.
• Cancer cells upregulate ribosome synthesis, a process that requires
large amount of phosphate
• The tumor suppressor p53 inhibits transcription of rRNA, in part
by inhibiting both RNA polymerase III. It may also inhibit the
production of mitochondrial rRNA.
Integrated tissue
Dominated by
cooperation
Cooperation within tumors
Angiogenesis
Fibroblast protease secretion
Cancer
Dominated by
confusion
Competition within tumors
O2, C–H, waste removal, space
Predation within tumors
Immune effector nastiness
Ecosystem
Dominated by
competition/predation
More Motivations
• Competition for phosphorus or any other resource
provided by the host potentially has enormous
significance to the clinical course of any
malignancy.
• The population of cancer cells within a given
tumor tends to be highly genetically and
physiologically varied.
• Natural selection, in addition to its role in
determining the incidence of cancer in the first
place, plays an important role in the clinical
behavior of cancer.
Objective
• Our objective is to incorporate natural
selection driven by competition for
resources, especially phosphorus, into a
mathematical model.
• The model tracks mass of healthy cells
within a host organ, mass of cancer cells of
various types and the number of blood
vessels within the tumor.
A mathematical model of tumor
growth with nutrient limitation:
Main assumptions
• The organ (x in kg), like the tumor (y in kg), is
capable of growth, but in the model's typical initial
state we assume that the organ is near some
genetically determined, ``healthy" carrying
capacity (k_h). z expresses mass (in kg) of tumor
microvessels.
• We assume that phosphorus content within the
organ is homeostatically regulated at a fixed value,
P (millimoles).