approximation of a cell cycle model

8/14/2019 Approximation of a Cell Cycle Model

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Approximation of a cell cycle modelin Budding Yeast

• Introduction

• Cell Cycle Model of Budding Yeast•

• Proposed Method

• Pre rocessin : Jacobian-based Local Refinement JLR

• Initialization of the Genetic Algorithm (GA)

• Results and Discussions

• Comparison between 2 Scenarios for Substitution of Excluded States

• Results of Substitution Excluded States with Mean Values

•

• Results of the GA with Parameter Estimation

• Conclusion and Future Works

Presented by Chaiyut Thanukaew 1



Introduction

• Molecular mechanisms in biological systems are complex and have dynamicinteractions.

• To express behaviors of a system, often a set of nonlinear ODEs is occupied.

• Since normally huge models, mathematical model reduction is used to gain insightinto the models and reduce computational cost.

• ,

model of budding yeast.

• The reduced model must retain essential features of the full model, traditionally,

criteria.

• For the cell cycle model of budding yeast, responses of the mass regardingexternal in ut is decided to be the reserved characteristic.

Objective: to minimize the number of states with preserving certain characteristicsof the model b usin enetic al orithm.

2



Cell Cycle Model of Budding Yeast (1)

4 hases of eukar otic* cell c cle; G1- S- G2-

and M-phase.

• G1 or Gap1 phase: the cell grows.

NuclearMigration

Formation

•copies of its chromosomes and get readyto divide.

SM

DNAReplication

ChromosomeSegregation,NuclearDivision

•

duplicated chromosomes.

• M or Mitosis phase: the cell separates theG1BudEmergence

the cell divides into 2 new cells

How to know which mechanisms of the cell cycleStart Growth

Spindle

Pole, Body

Duplication

Cytokinesis(Cell Division)

Figure 1: Cell Cycle of Budding Yeast

take place at what time?

CDK-Cyclin** is a protein active in regulatingthe cell cycle.

3* Eukaryote is a cell with visible nucleus e.g. yeast and mammal.** Cyclin Dependent Kinase and Cyclin



Cell Cycle Model of Budding Yeast (2)

Viability Criteria

• Correct sequences, i.e.Budding DNA synthesis Mitosis Chromosome

division.

• [mass] <= 10 all time.

Figure 2: Consensus Model of Budding Yeast

4



Constant Input Response (CIR) of Full Model (1)

CIR definition: “For each added constant input, the CIR is the amplitude and time of the mass state such that it becomes stable while, also, the whole model must comply the viability criteria ”.

c ppbdibd bdibd cscs PSicCdck C k V C k V Swik k dt

Sicd ]1].[14.[]5).[(]2).[(])5.["'(

]1[1,5,5,2,2,1,1,

added c pasas ... ,,,

Why adding at [Sic1]?

Added constant input

As Sic1 is a regular cyclin subunit called stoichiometric CDK inhibitor, directly addinga constant signal, physically, would mean putting some substances to boost up some

activities of the cell cycle, e.g. going faster into the G1-phase.

How to notice the response of the budding yeast model after adding up the constantsignal to the Sic1 state? The easiest answer is to measure how the mass physically reacts to this

5

perturbation.



Constant Input Response (CIR) of Full Model (2)

Response of the mass to different constant inputs

Mass Stable

added Time Amplitude

0.035 ~ 520 ~ 3

0.15 ~ 190 ~ 5

0.245 ~ 460 ~ 8

0.30 Inviable

6Figure 3: The mass state when different constant inputs are applied



Proposed Method (1)

We rewrite the full model as;The full model is written as;

Where; x : the vector of states

),,( ps x f x

),( p x f x

p : the vector of parameters, and

s : a vector where s i is “1” when the i-th state is included in the model“ ” .

Objective

The reduced model is defined such that its solution x* minimizes,,(*

s x x

the cost function , i.e.

)(min)( x x Ss

Where; : the cost function, and

7

: e se o a vec ors o eng f um er o s a es o e u mo e

whose entries are “0” or “1”.



Proposed Method (2)

Cost Function

The cost function is defined as;

1 X is not oscillator or inviable.

CIR

.1.1 X is oscillatory and viable.

Not count reduced models with negative cost as feasible solutions.

Where;

The relative model size is,

the number of states in the reduced model

the number of states in the full model f

r

N

N

8



Proposed Method (3)

Where (continued);

The CIR error is computed as the least square distance between one of the

full and one of the reduced model as following;

per N

i

RF RF i yi yi xi x1

22))()(())()((

per

i

F F i yi x1

22

)()(

• F F u wdifferent constant inputs are applied,

• N per is the maximum number of different constant inputs for which the,

• x R and y R represent the positions of the CIR for the reduced modelregarding different constant inputs added.

9



Generation “0” Generation “n”

Proposed Method (4)

Randomly generate anindividual

Do linear ranking of allindividuals

Copy 2 best individuals

from the previousgeneration (Elitism)

Re lacement

Jacobian-based localrefinement (JLR)

Calculate cost of eachindividual

Random selection 2parents acc. to their ranks

(till reachingsetting “n” or

Crossover and mutation

Total N i individuals

value or noimprovement

after some time)Jacobian-based local

refinement JLR

Calculate cost of each

offspringFigure 4: Flowchart of the GA

Total N i individuals

Applying to Reduce the Model

10



Jacobian matrix and JLR

Preprocessing : JLR (1)

The matrix of all first-order partial derivatives of a vector-valued function returningrelationship between equations in a system regarding that vector-valued function.

JLR uses information from Jacobian matrix for local refinement.

~

Considering the following equation;

ag

Where, J: the Jacobian matrix, and

.

If there is any row in such that all elements are zeros input-only state , and

If there is an column in such that all elements are zeros out ut-onl state .

~

J ~

Can leave out the input- and output-only states because of redundancy, e.g. the input-only state can be substituted by a constant and the output-only state can be indirectly

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o ta ne y some ot er states.



Preprocessing : JLR (2)Index table of the original cell cycle model

Input-only

states

Table 1: Index Table of the Cell Cycle Model 12Output-only

states

)(~

J diag J J



Initialization of the GA (1)

Problem: Very hard to obtain individuals being oscillatory and viable in the gen. “0”.

Reasons would be as following;

•causing that half of the number of bits is “0” in average (18 from 36 bits). There isthe problem, e.g. assume that oscillatory individuals must comprise of at least 26states, thus it is hard to obtain this number.

• The generated individuals lack of some crucial states, e.g. mass and some states ofcyclins. Thus, leaving out only one of these states causes the reduced models non-oscillatory or inviable.

Solution introduce preconditions to generates the individuals in the initial generation

of the GA.

13



Initialization of the GA (3)Example: Feasible individual generated in gen. “0” with preconditions

1

R2

1

bit

bit State 1: set to “1” (mass)

State 3 and 4: randomly generated

1

1

5

4

bit

bit State 4: set to “1” (viability concerned)

State 5: set to “1” (constant input added)

L26bit State 26, 27: likely set to “1s” (only missing

one s a e, non-osc a ory

1

1

35

34

bit

bit State 33, 34, 35 and 36: set to “1s”(viability concerned)

14Figure 5: Example of Generating a Feasible Individual in Gen. “0”



Comparison between Zero VS Mean-valued Sub.

2 scenarios of substitution an excluded state;

• Substitution with zero: the excluded states is completely cut away, and

• Substitution with mean value: the excluded state still has influence, but not.

Case #1 Case #2 Case #3

Population Size 80 100 80

Fitness Assignment Linear ranking Linear ranking Linear ranking

Selection Fitness-based Fitness-based Fitness-based

Crossover Uniform Uniform 2-point

Mutation Bit mutation prob.=0.1 Bit mutation prob.=0.1 Bit mutation prob.=0.1

Replacement Scheme Whole population

replacement with elitist

Whole population


Whole population


15Table 2: Parameter Setting of 2 Substitution Scenarios

no.= no.= no.=



Results : Comparison between Zero VS Mean-valued Sub.

The best result regarding Table 3 comes from the mean-valued substitution

2. Reduced model with mean-valued

substitution

Case #1 Case #2 Case #3

Total State No. 23 24 25

Total Cost 0.7244 0.7521 0.7754

Table 3: Results of Mean-valued Substitution

u w -v u u u usimulations since it returns the better cost value (the lower value).

The total state number can be subtracted 3 because the states 34, 35, 36 areoutput-only states added to determine the viability (see slide 12).

Not obvious seen difference between each case of the same substitution

16

.



Results : Mean-valued Substitution (1)

Table 4: Results of Mean-valuedSubstitution from Several Simulations

Purpose: to prove that the results

• The best reduced models “S1” and“S2” contains 23 states.

.

• Although, different number ofstates for each result, all results aresomehow in the same pattern (blue

.

• If fixing highlight areas, the best

result can be obtained from varying-(search space is reduced).

• Further test “S1” and “S2”

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the full model .



Results : Mean-valued Substitution (2)The cell cycle of the full model VS the cell cycle of the reduced model

18

gure : e yc e o e u o e e yc e o n a e

Cell cycle of “S2” is very similar to “S1” except that Clb2T is higher (max. nearly 1.5).



Results : Mean-valued Substitution (4)Reduced model “S1” as the Jacobian index table

: Excluded-

dinamic)

: Substitutedby constant

: Non-zeroJacobian

Table 5: Reduced Model “S1” Substituting in the Jacobian Index Table19



Results : Mean-valued Substitution (4)Reduced model “S1” as the Jacobian index table

Table 6: Jacobian Index Table of the Reduced Model “S1”

20Note: State 24 (Tem1) is input-only state but cannot be left out since itinvolves resetting parameters.



GA with Parameter Estimation (1)

Idea: number of feasible solutions mi ht be increased if the thresholds of determination the viability criteria are changed.

Implementation method: to put 4 more bits representing real value numbers used as the.

RepresentationRepresentation 36 Binary bit strings and 4 positive real values

Po ulation sizePo ulation size 80

Fitness assignmentFitness assignment Linear ranking

SelectionSelection Fitness-based

CrossoverCrossover Uniform crossover with prob. = 0.5 (crossover in the parameter part is

substituted by the median values of the parents)

MutationMutation Bit mutation with prob. = 0.1 and normal distribution added with theprevious value in the parameter part

ReplacementReplacementSchemeScheme

Whole population replacement with elitist no. = 2

21Table 7: Parameter Setting of the GA with Parameter Estimation



Results: GA with Parameter Estimation

Results of the GA with Parameter Estimation

Reduced Model with

Parameter Estimation

Discussions

The results are not better than the results with no

Round#3Ind.1

Round#3Ind.2

Kez (0.3) 0.4572 0.4242

parameter estimation. The reasons would be asfollowing;

• Ina ro riate estimation of “Thres. 1”

Kez2 (0.2) 0.1720 0.2485

Thres. 0.1 0.1050 0.0806

because it is used as the threshold for 3 states[ORI], [BUD] and [SPN],

• Newl estimated arameters cannot leadThres. 1 0.6481 0.9425

Total State

No.

24 25

feasible individuals into optimal region in thesearch space, and

• New arameters are enerated b the normal

Total Cost 0.7765 0.7954 distribution with a small variance.

a e : e es esu o w arame er

Estimation (from 3 times of simulation) 22



Conclusion and Further Works

• The best result contains 23 states. 3 more states can be ignored because of theoutput-only states.

• Appropriate GA’s parameters might increase efficiency of the simulation, e.g. lessenthe com utational time, ex lore the search s ace intelli entl .

• The GA hybridizes with the Jacobian-based local refinement can reduceunnecessary computation, e.g. the GA with JLR in this work.

• Although GA can be applied to resolve a given problem without prior knowledge ofsuch that problem, initialization of the GA is applied to this problem because ofphysical reality and model constraints.

• Further works would be to study more about the GA with parameter estimation and tocombine other strategies of model reduction to the GA, e.g. to lump 2 continuous states

together.

1 2 3 1 2 3

23Thank You



Appendix A : CIR of the Full Model

.that Kadded is increased 0.01 eachtime (from 0 to 0.28)

.that Kadded is the median values ofeach group

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approximation of a cell cycle model

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