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LAST TIME. Linear Equations General Form: x n +1 = ax n + b If b = 0, the equation is homogeneous If b 0, the equation is inhomogenous Equilibrium , x e , is achieved if x n +1 = x n = x e . Linear discrete models have a single unique equilibrium if a , is not 1 .
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LAST TIME • 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 ifxn+1 = xn = xe. • Linear discrete models have a single unique equilibrium ifa, 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 stableif • 1. Successive iterations of the model approach the equilibrium. • 2. The slopea is less than 1.
Systems of Linear Difference Equations • Sometimes you will be interested in two or more quantities that influence each others change from generation to generation.
Systems of Linear Difference Equations • Any system of linear first order difference equations can be converted to a single higher order system. • Increase order of one of the the equations, say the x equation 2. Eliminate yn+1
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
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.
Finding the Solution • Look for solutions of the form: xn = Cn • Substitute into to get divide Cn by to obtain the Characteristic Equation
Finding the Solution • Solutions of the characteristic equation are called eigenvalues. • The properties of the eigenvalues uniquely determine the behavior of the solutions.
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: For real, distinct evals: For real, equal evals:
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
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
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
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
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, r a b
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
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:
Complex Eigenvalues • So Therefore But this is a complex function …
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.
Example Solve: Characteristic Equation: Eigenvalues: Solution:
Failure of Programmed Cell Death and Differentiation as Causes of Tumors Some simple mathematical models
Adenomatous polyps: 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. 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 • stem cell • differentiated cell Not considered! 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
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 modelsTomlinson 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 P1 P0 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
Parameters a1 = probability of stem cell (P0) death a2 = probability of stem cell (P0) differentiation a3 = probability of stem cell (P0) renewal b1 = probability of progenitor cell (P1) death b2 = probability of progenitor cell (P1) differentiation b3 = probability of progenitor cell (P1) renewal g = 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 Cells must do one of three things!
P0 P1 P2 P3 C S F D t0 t1 t2 Model Schematic Stage of Differentiation Number of Cells Generation Time
Normal Cell Division Model Equation At Equilibrium Stem Cell Population, C There is a unique probability of proliferation at which the stem cell population exactly renews itself. If 3 rises above or falls below 1/2, Cn Increase or decreases exponentially.
Normal Cell Division Model Equation Semi-differentiated (Progenitor) Cell Population, S At Equilibrium
Model Equation: Homogeneous Solution: Semi-differentiated (Progenitor) Cell Population, S Particular Solution: Let To Find:
Model Equation: General Solution: Semi-differentiated (Progenitor) Cell Population, S Apply Initial Condition: To Find:
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 Semi-differentiated (Progenitor) Cell Population, S
Normal Cell Division Model Equation At Equilibrium Fully-differentiated Cell Population, F • 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 Tissue Composition 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 t2is 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