• Linear Equations– General Form: xn+1 =axn+ b

• If b = 0, the equation is homogeneous• If b 0, the equation is inhomogenous

• Equilibrium ,xe, is achieved if xn+1 = xn = xe.

– Linear discrete models have a single unique equilibrium if a, is not 1.

• If a = 1 , then either there are no equilibria or all points are equilibria ( b=0).

• Stability: An equilibrium of a linear discrete model is stable if– 1. Successive iterations of the model approach the

equilibrium. – 2. The slope a is less than 1 .

€

≠

LAST TIME

Systems of Linear Difference Equations

• Sometimes you will be interested in two or more quantities that influence each others change from generation to generation.

€

xn +1 = a11xn + a12yn

€

yn +1 = a12xn + a22yn

Systems of Linear Difference Equations

• Any system of linear first order difference equations can be converted to a single higher order system.

1. Increase order of one of thethe equations, say the x equation

2. Eliminate yn+1

€

xn +1 = a11xn + a12yn

€

yn +1 = a12xn + a22yn

€

xn +2 = a11xn +1 + a12yn +1

€

xn +2 = a11xn +1 + a12(a21xn + a22yn )

Systems of Linear Difference Equations

• Any system of linear first order difference equations can be converted to a single higher order system.

3. Eliminate yn

€

xn +1 = a11xn + a12yn

€

yn +1 = a12xn + a22yn

€

yn = xn +1 − a11xn

€

xn +2 = a11xn +1 + a12(a21xn + a22yn )

Systems of Linear Difference Equations

• Any system of linear first order difference equations can be converted to a single higher order system.

This equation is 2nd order and requires two previous data points in order to determine the future value of x.

€

xn +2 − (a11 + a22)xn +1 + (a12a21 + a22a11)xn = 0€

xn +1 = a11xn + a12yn

€

yn +1 = a12xn + a22yn

Finding the Solution

• Look for solutions of the form: xn = Cn

• Substitute into

to get

divide Cn by to obtain the Characteristic EquationCharacteristic Equation

€

xn +2 − (a11 + a22)xn +1 + (a12a21 + a22a11)xn = 0

€

Cλn +2 − (a11 + a22)Cλn +1 + (a12a21 + a22a11)Cλn = 0

€

2 − (a11 + a22)λ + (a12a21 + a22a11) = 0

Finding the Solution

• Solutions of the characteristic equation are called eigenvalues.

• The properties of the eigenvalues uniquely determine the behavior of the solutions.

€

1,2 =a11 + a22 ± (a11 + a22)2 − 4(a11a22 − a12a21)

2

Principle of Superposition

• For linear difference equations; if several different solutions are known, then any linear combination of the these solutions is again a solution.

• Therefore the General Solution is:

€

xn = A1λ1n + A2λ 2

n , For real, distinct evals:

€

1 ≠ λ 2

€

xn = A1λn + A2nλn, For real, equal

evals:

€

1 = λ 2 = λ

Dominant Eigenvalue

• The dominant eigenvalue is the one with largest magnitude, ie the largest absolute value.

• Because solutions to second order discrete equations are of the form:

the dominant eigenvalue will have the strongest effect on the behavior of the solutions€

xn = A1λ1n + A2λ 2

n ,

Example

General Form of 2nd Order Discrete Equations

• When b = 0, the solution for real, distinct eigenvalues is

• When b = constant, the solution for real distinct eigenvalues is

€

xn +2 + a1xn +1 + a2xn = bn

€

xn = A1λ1n + A2λ 2

n ,

€

xn = A1λ1n + A2λ 2

n + K

Example

Qualitative Behavior of Linear,Discrete Equations

• An mth order, linear discrete (difference) equation takes the form

• The order, m, refers to the number of pervious generations that directly impact the value of x in a given generation

• When coefficients are constants and bn = 0, the equation is homogeneous and solutions are linear combinations of the form: Cn

€

xn +m + a1xn +m−1 + a2xn +m−2 + ...+ am xn = bn

Qualitative Behavior of Linear,Discrete Equations

• The number of basic solutions to a linear, discrete equation is determined by its order.– In general, an mth order equation has m basic solutions

• The General Solution is a linear combination of the basic solutions (provided all values of the eigenvalues are distinct)

• The eigenvalue with the largest magnitude will have the strongest effect on the behavior of the solutions

QUESTIONWhat if the Eigenvalues are

Comnplex Numbers?

Complex Eigenvalues

• The solution to a general characteristic polynomial can be a complex number.

• A complex number, a + bi, is the point in the complex plane with coordinates (a,b).

• Or equivalently,

€

a = rcosφ

€

b = rsinφ

€

r = a2 + b2

€

φ=arctanb

a

⎛

⎝ ⎜

⎞

⎠ ⎟ a

b

r

Complex Eigenvalues

• Complex e-vals occur in conjugate pairs, for example: • The general solution will then be:

• What is (a +bi)n?• Recall Euler’s Formula

a + bi = r(cos + isin) = rei

a - bi = r(cos - isin) = re-i

€

1 = a + bi

€

2 = a − bi

€

xn = A1(a + bi)n + A2(a − bi)n

Complex Eigenvalues• Using Euler’s Formula: (a +bi)n = (rein = rnein

(a + bi)n = rn[cos(n + isin(n]

• Similarly: (a - bi)n = (re-in = rne-in

(a - bi)n = rn[cos(n - isin(n]

Now substitute this into:

€

xn = A1λ1n + A2λ 2

n ,

€

xn = A1(a + bi)n + A2(a − bi)n

Complex Eigenvalues• So

Therefore

But this is a complex function …

€

xn = A1(a + bi)n + A2(a − bi)n

€

=A1rn (cosnφ + isin nφ) + A2r

n (cosnφ − isin nφ)

€

xn = B1rn cosnφ + iB2r

n sinnφ

Complex Eigenvalues

• Define a real-valued solution by the superposition of the real and imaginary parts:

• Therefore complex eigenvalues are associated with oscillatory solutions. The amplitude grows if r > 1, decreases if r < 1, and remains constant if r = 1.

• Periodic solutions occur if is a rational multiple of and r = 1.

€

xn = C1rn cosnφ + C2r

n sin nφ

Example

€

xn +2 − 2xn +1 + 2xn = 0

€

2 − 2λ + 2 = 0

€

=1± i

€

a = b =1

€

r = a2 + b2 = 2

€

φ=arctan(b

a) =

π

4

€

xn = C1 2n(cos

nπ

4+ C2 sin

nπ

4)

Solve:

Characteristic Equation:

Eigenvalues:

Solution:

Failure of Programmed Cell Death and Differentiation as

Causes of Tumors

Some simple mathematical models

Hallmark Cancer Capabilities

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

?

Benign versus Malignant• Benign tumors are generally

– composed of well-differentiated, slow growing cells; – enclosed in a fibrous capsule; – relatively innocuous,

• Malignant tumors are generally – composed of poorly-

differentiated, rapidly proliferating cells;

– invasive and destructive to normal tissue;

– metastatic, or capable of spreading to other sites of the body.  

Adenomatouspolyps:

Malignant gastric carcinoma

Cancer Stem Cell Hypothesis

• Cancer stem cells have been identified in malignancies of the breast, brain, and blood and are believed to drive disease progression in these and possibly most cancers.

Hierarchal Cellular Systems

• Stem Cells - the most naive

• Progenitor Cells - precursors for mature cells

• Differentiated Cells - carry out specific functions

Types of Division in Model

Symmetric self-renewal

A new stem cell is added

Asymmetric self-renewal

Number of stem cells stays the same

One differentiated cell is added

Non-self-renewal division

One stem cell is removed

Two differentiated cells are added

- stem cell

- differentiated cell

Not considered!

Programmed Cell Death

• A normal physiological response to cell stress, cell damage or conflicting cell division signals

• Many cancers are hypothesized to arise from and are difficult to eradicate due to the failure to respond to apoptotic signals

Role of PCD in Tumorigenesis

• Precise role is still unclear• Failure of PCD might give cells the equivalent of

a replicative advantage– Failure to die is effectively the same as more rapid cell

division

• Failure of PCD may lead to an increase in the intrinsic mutation rate– Cells live longer and are exposed to more mutagens or

acquire more spontaneous mutations

Failure of programmed cell death and differentiation as causes of tumors: Some simple mathematical models

Tomlinson and Bodmer, PNAS 1995

Basic Models of Tumor Growth

• Assume tumor grows by increased cell division• A mutant cell population increases as mn+1 = 2mn mn = m02n

• If mutants have a replicative advantage mn = m0[2(1+w)]n, where w is the selective advantage of the mutant relative to a mean population of normal cells

• All descendants of the original mutant cell population behave the same way

What happens when PCD is included?

• This model doesn’t work• It cannot be assumed that cells behave as their parents

do• A mutation occurring in a stem cell may have no

effect until it is fully differentiated and about to undergo PCD– This cell may have divided many times – When cells differentiate and die a planned death, the effect

of mutations will vary depending on when and where they occur

• Timing is crucial!

Goal of the Study

• Set up a simple mathematical model of tumorigenesis by failure of PCD and failure of differentiation

• Use the model to demonstrate how tumor growth proceeds under these circumstances

• Compare these results to the exponential growth predicted by increased cell division models

Definitions

• P0 = a self-renewing population of stem cells

• P1 = a population of cells at an intermediate differentiation state-- progenitor cells

• P2 = a population of fully differentiated cells

• P3 = the dead cell population

P0

P1

P2

P3

Variables

• Cn = number of stem cells after n divisions

• Sn = number of intermediate cells after n divisions

• Fn = number of fully differentiated cells after n divisions

Built in Assumptions

• The number of cells in Pn (n = 0,1,2) depends on – The number of cells in Pn-1,for n = 1,2– The rate of division of cells in Pn-1, for n = 1,2– The probability that cells in Pn-1 differentiate into

Pn cells rather than remain Pn-1 or die, for n = 1,2– The rate of division of Pn cells, for n = 0,1– The probability that cells in Pn differentiate into

Pn+1 cells or die rather than remain in Pn, for n = 0,1

Parameters1 = probability of stem cell (P0) death

2 = probability of stem cell (P0) differentiation

3 = probability of stem cell (P0) renewal

1 = probability of progenitor cell (P1) death

2 = probability of progenitor cell (P1) differentiation

3 = probability of progenitor cell (P1) renewal

= probability of mature cell (P2) death

t0,t1, = time for one cell division to occur for stem cells (P0) and progenitor cells (P1) respectively.

Constraints

€

1α + 2α + 3α =1

€

1β + 2β + 3β =1

€

iα , iβ > 0

Cells must do one of three things!

Model SchematicStage of Differentiation

Number ofCells

GenerationTime

C S F D

€

1

€

2

€

3

€

1

€

2

€

3

€

P0 P1 P2 P3

t0 t1 t2

Normal Cell DivisionStem Cell Population, C

€

Cn + 1 = 2α 3Cn

€

Ce = 2α 3Ce → α 3 = 12,Ce = C0

There is a unique probability of proliferation at which the stem cell population exactly renewsitself. If 3 rises above or falls below 1/2, Cn Increase or decreases exponentially.

Model Equation

At Equilibrium

€

Cn = C0(2α 3)n

Normal Cell DivisionSemi-differentiated (Progenitor) Cell

Population, S

€

Sn + 1 = 2β 3( 0t1t)Sn + 2α 2Cn

€

Se = 2β 3( 0t1t)Se + 2α 2Ce → Se =

2α 2Ce

1− 2β 3( 0t1t)

Model Equation

At Equilibrium

Semi-differentiated (Progenitor) Cell Population, S

€

Sn + 1 = 2β 3( 0t1t)Sn + 2α 2C0 2α 3( )

n

Model Equation:

Homogeneous Solution:

€

Sn = A1 2β 3( 0t1t)( )

n

Particular Solution: Let

To Find:

€

Sn = A2 2α 3( )n

€

A2

=α

2C0

α3

− β3

0t1t( )

Semi-differentiated (Progenitor) Cell Population, S

€

Sn + 1 = 2β 3( 0t1t)Sn + 2α 2C0 2α 3( )

n

Model Equation:

General Solution:

€

Sn = A1 2β 30t1t( )

n

+α

2C0

α3

− β3

0t1t

2 α 3( )n

Apply Initial Condition:

To Find:

€

A1

= S0 −α

2C0

α3

− β3

0t1t( )

Semi-differentiated (Progenitor) Cell Population, S

€

Se = 2β 3( 0t1t)Se + 2α 2Ce → Se =

2α 2Ce

1− 2β 3( 0t1t)

At Equilibrium

•Case 1: There is no realistic equilibrium point if

23t0/t1 > 1 because when t1/t0 < 23 (ie when the

cell cycle time for P1 relative to P0 is less than twice the probability of renewal), then Se is negative

In this case: Sn increases exponentially

•Case 2: There is no equilibrium if Cn is not in equilibrium• Sn behaves as Cn does

Normal Cell DivisionFully-differentiated Cell Population, F

€

Fn + 1 = 2β 2( 0t1t)Sn + [1− γ( 0t

t 2)]Fn

€

Fe = 2β 2( 0t1t)Se + [1− γ( 0t

t 2)]Fe → Fe =

2β 2( 0t1t)Se

γ 0t2t

•Case 1: When Cn and Sn are in equilibrium, so is Fn

•Case 2: There is not equilibrium if Sn is not in equilibrium

Fn behaves as Sn does

Model Equation

At Equilibrium

Model Predictions

Model Tissue Composition

Stem CellsProgenitor CellsMature Cells

1.2%6.2%

92.6%

About These Results

• Results illustrate the increased complexity of behavior that accompanies models that considers cell differentiation and PCD

• If we restrain parameters so that the cell populations are in equilibrium, the limits for the stem cell population are restrictive, but restrictions weaken for the other cell populations

• Now let’s analyze the case in which a mutation has altered the proportions of cells dying, differentiating or renewing themselves in order to determine the effects on tumorigenesis

Changes in the Probability of F-Cells Undergoing PCD,

• What happens if changes by where 0 < + < 1?

• This mutation might have occurred in the P2 population itself and if so would not have had a large effect.

• It is more likely that the mutation occurred in P0 or P1, but only has an effect on the P2 cells.

Probability of Death/Survival

is the probability of a fully differentiated cell dying

• t2 is the time it for a takes a cell to die

t0 t2 is the probability that a mature cell dies in the time it take for a stem cell to divide.

• A fully mature cell either lives or dies

• The probability of survival is one minus the probability of death during the time it takes a stem cell to divide ie 1 - t0 t2

After the Mutation

• Therefore a mutation in the probability of PCD does not lead to a exponential tumor growth, it simply leads to a new equilibrium state.

• If < 0 the new steady state is larger• If > 0 the new steady state is smaller• If ~ 0 then new equilibrium can be very large a benign tumor

• Cn and Sn are always unchanged.

€

Fn + 1 = 2β 2( 0t1t)Sn + [1− (γ + δ )( 0t

t 2)]Fn

€

Fe = 2β 2[ 2t1(γ+δ)t

]Se

Mutation Leads to Benign Growth

500% increase in cell number after 25 generations.

Late Stage Mutation

Dependence of Fe on

Changes in the Proportion of S-cells undergoing PCD, 1

• What happens if a mutation occurs that causes 1 to be reduced by an amount ?

• Cells that fail to die are partitioned between 2 and 3 relative to their normal values

After Mutation

€

Sn + 1 = 2β 3[1 +δ

(β 2 + β 3)]Sn(t 0

t1) + 2α 2Cn

€

Se =2α 2Ce

1 − 2β 3[1+δ

(β 2 + β 3)](t 0

t1)

€

>1 − 2β 3(β 2 + β 3) t 0

t1 ⎛ ⎝ ⎜ ⎞

⎠ ⎟

2β 3 t 0t1

⎛ ⎝ ⎜ ⎞

⎠ ⎟

No SteadyState if

New condition

Mutation Leads to Benign Growth

New Tissue Resembles Original Tissue

Stem CellsProgenitor CellsMature Cells

4.8%1.2%

95%

160% increase in cell number after 25 generations.

Mutation Leads to Explosive Growth

Tissue Composition Changing

Stem CellsProgenitor CellsMature Cells

13%0.04%

86.96%

Huge increase in cell number after 15 generations.

Interpretation• If is too large, cells in the P1 population undergo

exponential growth.• Recall that for normal cell growth the condition for

equilibrium of S-cells was 23t0/t1 < 1 , therefore the tendency towards non-equilibrium is made more likely by the term (/( 3))

• PCD does not necessarily lead to exponential growth, a new higher equilibrium may be reached.

• For F-cells the existence of an equilibrium depends solely on whether or not the S-cells are in equilibrium.

• C-cells are unaffected.

Changes in the Proportion of C-cells Undergoing PCD, 1

• What happens if a mutation occurs that causes 1 to be reduced by an amount ?

• Cells that fail to die are partitioned between 2 and 3 relative to their normal values

After Mutation

€

Cn + 1 = 2α 3[1+δ

(α 2 + α 3)]Cn

€

=( 1

2α 3− 1)(α 2 + α 3)

Equilibriumoccurs if andonly if

Note: if 3 = 1/2, then =0 is theonly possibility for equilibrium.

Mutation Leads to Explosive Growth

Shift in Stem Cell Fraction

Stem CellsProgenitor CellsMature Cells

8.7%3.9%

87.6%

Huge increase in cell number after 15 generations.

Summary

• When stem cells fail to die or fail to differentiate, exponential growth in cell number always occurs.

• When semi-differentiated cells fail to die or differentiate, exponential growth does not always occur.

• When PCD of fully differentiated cells fails to occur, there is never exponential growth in cell numbers.

Changes in the Proportion of Stem cells Undergoing

Differentiation , 2

• What happens if a mutation occurs that causes 2 to be reduced by an amount ?

• Cells that do not die are forced to symmetrically self-renew. Therefore 3 is increased by an amount

After Mutation

You Tell Me What Happens?

After Mutation

There is no steady state and explosive growth of stem cells occurs. This leadsto explosive growth of all populations.

Overall, this result is very similar to what happens when stem cells fail to die.

€

Cn + 1 = 2α 3[1+ δ]Cn

Changes in the Proportion of Progenitor Cells Undergoing

Differentiation, 2

• What happens if a mutation occurs that causes 2 to be reduced by an amount ?

• Cells that do not die are forced to symmetrically self-renew. Therefore 3 is increased by an amount

After Mutation

You Tell Me What Happens?

After Mutation

€

Sn + 1 = 2β 3[1+ δ]Sn(t 0t1

) + 2α 2Cn

€

Se =2α 2Ce

1− 2β 3[1+ δ](t 0t1

)

€

>1− 2β 3 t 0

t1 ⎛ ⎝ ⎜ ⎞

⎠ ⎟

2β 3 t 0t1

⎛ ⎝ ⎜ ⎞

⎠ ⎟

No Steady State If:

This result is very similar to what happens when stem cells fail to die, however the threshold for in

this case is larger.

Conclusion

• When PCD fails in fully differentiated cells, there is never explosive growth in cell numbers.

• The failure of PCD or differentiation in progenitor is sometimes sufficient buy does not always lead to tumorigenesis.

• The failure of PCD or differentiation occurs in stem cells, the result is always explosive growth.

• Mutations in stem cells are a powerful mechanism for tumorigenesis.