Biological Stoichiometry ofTumor 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 DMS-

0077790

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 hard-edged 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.

0

1000

2000

3000

4000

5000

6000mg C

10 20 30 40 50 60days

High (Light/Nutrient), K=6000 mgC, P=100 mgP

0

2000

4000

6000

8000

10000

12000mg C

10 20 30 40 50 60 70 80 90days

Very High (Light/Nutrient), K=12000 mgC, P=100 mgP

x0 K

y

y

x0 K

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.

A

B

(________) prey

(- - - - - - -) predator

Region II

Region II

Region I

Region I

Tumor

The Growth Rate Hypothesis

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.

Body C:N:P

food quality constraints on growth /

reproduction

resource competition

trophic efficiency

nutrient recycling

natural selection on growth rate

cellular investment (ribosome content)

biochemical investment (RNA:protein)

The first picture of a ribosome. Cate et al. (1999) Science 285: 2095-2104.

Are P-rich Animals Also RNA-rich Animals?

0.40.2 0.6 0.8 1.0 1.2 1.40

0

0.5

1.0

1.5

2.0

2.5

%P

in

th

e B

od

y

% RNA-P in the Body

Scapholebris mucronata

Daphnia magna (small)

Daphnia magna (large)

Bosmina longirostris

Y = 1.2X + 0.66 p = 0.02 R-sq = 0.96

0

0.5

1.0

1.5

2.0

2.5p < 0.001

ABA

A

B

%P

in

th

e B

od

y

Bosmina longirostris

Daphnia magna (large)

Daphnia magna (small)

Scapholebris mucronata

A

AA

Bp < 0.001

% R

NA

in

th

e B

od

y

0

4

8

12

16

From: Dobberfuhl, D.R. 1999. Elemental stoichiometry in crustacean zooplankton: phylogenetic patterns, physiological

mechanisms, and ecological consequences. PhD dissertation, Arizona State University, Tempe, AZ.

1:1

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 tissueIntegrated tissue Dominated by Dominated by cooperationcooperation

EcosystemEcosystem Dominated by Dominated by

competition/predationcompetition/predation

CancerCancer Dominated by Dominated by

confusionconfusion

Cooperation within tumorsCooperation within tumors

AngiogenesisAngiogenesisFibroblast protease secretionFibroblast protease secretion

Competition within tumorsCompetition within tumorsOO22, C–H, waste removal, space, C–H, waste removal, space

Predation within tumorsPredation within tumorsImmune effector nastinessImmune effector nastiness

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).