prism - win.tue.nlevink/education/2if35/pdf/2if35-05a.pdf12 / department of mathematics and computer...

12

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PRISM

2IF35 – week 5

12


properties of discrete-time Markov chains

I until:

P ≥ 0.92(

¬illegal U goal

)I bounded until:

P≥ 0.92(

¬illegal U≤137 goal

)

I eventually:

P≥ 0.92(

true U

=137

goal

)

≡

P≥ 0.92(

♦

=137

goal

)

I always:

P≥ 0.92(

� goal

)

≡

P≤ 0.08(

¬♦¬goal

)

I nested:

P≥ 0.92(

¬illegal U

P≥ 0.9999

(�

=13

goal

)

)

12



I until:

P≥ 0.92(

¬illegal U goal

)

I bounded until:

P≥ 0.92(

¬illegal U≤137 goal

)

I eventually:

P≥ 0.92(

true U=137 goal

)

≡

P≥ 0.92(

♦=137 goal

)

I always:

P≥ 0.92(

� goal

)

≡

P≤ 0.08(

¬♦¬goal

)

I nested:

P≥ 0.92(

¬illegal U

P≥ 0.9999

(�=13goal

)

)

12



I until: P≥ 0.92(¬illegal U goal

)I bounded until: P≥ 0.92

(¬illegal U≤137 goal

)I eventually: P≥ 0.92

(true U=137 goal

)≡ P≥ 0.92

(♦=137 goal

)I always: P≥ 0.92

(� goal

)≡ P≤ 0.08

(¬♦¬goal

)I nested: P≥ 0.92

(¬illegal U P≥ 0.9999

(�=13goal

) )

12


properties of continuous-time Markov chains

I P≥ 0.92(¬illegal U≤133.4 goal

)I P≥ 0.92

(¬illegal U[61.1,133.4] goal

)

12


Prism: a probabilistic modelchecker

I discrete-time and continuous-time Markov chainsI reactive module-type of modelling languageI specifications in PCTL and CSLI see http://www.prismmodelchecker.org

Kwiatkowska et al., Proc. QEST (2004)

http://www.prismmodelchecker.org

12


a kitchen-salt example

Na++ Cl−

K fKb

NaCl

12


a Prism model

module Solution

Na=0, Cl=0, NaCl=50;

[ ] Na > 0 & Cl > 0 -> Kf∗Na∗Cl :

Na’=Na-1 & Cl’=Cl-1 & NaCl’=NaCl+1;

[ ] NaCl > 0 -> Kb∗NaCl :

Na’=Na+1 & Cl’=Cl+1 & NaCl’=NaCl-1;

endmodule

12


experiments

percentage NaCl

12


CSL properties

I ’init’ ⇒ �0.98 (Na = Cl ) (valid for Kf=0.01,Kb=1)

levels of Na and Cl are always the same

I ’init’ ⇒ ♦ (Na > NaCl ) (valid for Kf=0.01,Kb=1)

at some point in time the Na level will exceed that of NaCl

I ’init’ ⇒ ♦≤10 (Na > NaCl ) (not valid for Kf=0.01,Kb=1)

the same but within 10 time units

I ’init’ ⇒ ♦≤20 (Na > NaCl ) (valid for Kf=0.01,Kb=1)

again the same but now within 20 time units

12


Example: two process mutual exclusion

nondeterministic

module M1x : [0..2] init 0;

[] x=0 -> 0.8:(x’=0) + 0.2:(x’=1);[] x=1 & y!=2 -> (x’=2);[] x=2 -> 0.5:(x’=2) + 0.5:(x’=0);

endmodule

module M2y : [0..2] init 0;

[] y=0 -> 0.8:(y’=0) + 0.2:(y’=1);[] y=1 & x!=2 -> (y’=2);[] y=2 -> 0.5:(y’=2) + 0.5:(y’=0);

endmodule

12


Example: N-place queue and server

ctmc

const int N = 10;const double mu = 1/10, lambda = 1/2, gamma = 1/3;

module queueq : [0..N];

[] q<N -> mu:(q’=q+1);[] q=N -> mu:(q’=q);[serve] q>0 -> lambda:(q’=q-1);

endmodule

module servers : [0..1];

[serve] s=0 -> 1:(s’=1);[] s=1 -> gamma:(s’=0);

endmodule

12


circadian clock: activator and repressor

12


circadian clock

ctmcconst int N=1000000;const double k;

module activator_gene

da : [0..1] init 1;da_a : [0..1];

[transc_da] da=1 -> da : true;[transc_da_a] da_a=1 -> da_a : true;[bind_a] da=1 & da_a=0 -> da : (da’=0) & (da_a’=1);[rel_a] da=0 & da_a=1 -> da_a : (da’=1) & (da_a’=0);

endmodule

12


activator modules

module activator_mRNAma : [0..N]; // amount of activator mRNA

[transc_da] ma<N -> ma’=ma+1;[transc_da_a] ma<N -> ma’=ma+1;[transl_a] ma>0 -> ma : true;[deg_ma] ma>0 -> ma : ma’=ma-1;

endmodule

module activator_proteina : [0..N]; // amount of activator protein

[transl_a] a<N -> a’=a+1;[bind_a] a>0 -> a : a’=a-1;[bind_r] a>0 -> a : a’=a-1;[rel_a] a<N -> a’=a+1;[rel_r] a<N -> a’=a+1;[deg_a] a>0 -> a : a’=a-1;[deactive] a>0 -> a : a’=a-1;

endmodule

12


repressor modules// rename activator gene modulemodule repressor_gene =

activator_gene[da=dr, da_a=dr_a, bind_a=bind_r, rel_a=rel_r,transc_da=transc_dr, transc_da_a=transc_dr_a]

endmodule

// rename activator mRNA modulemodule repressor_mRNA =

activator_mRNA[ma=mr, transl_a=transl_r, transc_da=transc_dr,transc_da_a=transc_dr_a, deg_ma=deg_mr]

endmodule

module repressor_proteinr : [0..N];

[transl_r] r<N -> (r’=r+1);[deg_r] r>0 -> r : (r’=r-1);[deactive] r>0 -> r : (r’=r-1);[deg_c] r<N -> (r’=r+1);

endmodule

12


complexes and rates

module inactive_proteinc : [0..N]; // amount of inactivated protein

[deactive] c<N -> (c’=c+1);[deg_c] c>0 -> c : (c’=c-1);

endmodule

module rates[transc_da] true -> k*50 : true;[transc_da_a] true -> k*500 : true;[transc_dr] true -> k*0.01 : true;[transc_dr_a] true -> k*50 : true;[transl_a] true -> 50 : true;[transl_r] true -> 5 : true;[bind_a] true -> 1 : true;[bind_r] true -> 1 : true;[deactive] true -> 2 : true;...

endmodule

12


reward structure

const double T;

rewards "activated_protein"true : a;endrewards

12


simulation

single run

average over 20 runs

number of activator proteins over time

12


simulation

single run average over 20 runs

number of activator proteins over time

12


transcription factor

activator level

12


aa-tRNA competition

I Fluitt, Pienaar and ViljoenComputational Biology and Chemistry 31 (2007) 335–346

I building on work of Rodnina c.s.codon CUC with Leu cognate and Phe near-cognate

I simplifications:• constant tRNA pool• no ribosome-ribosome interaction• constants not measured are not rate limiting

I simulation effort: correlation with competition

12


transcription & translation

protein

mRNA

DNA

mRNA: string of codons protein: string of amino acids

12


the genetic code

12


codon & anticodon

anticodon

aminoacid

tRNA

codonmRNA

E. coli: 64 codons, 48 aa-tRNAs/anticodons, 20 amino acids

12


aa-tRNA

specifity aa-tRNA by anticodon

12


ribosome translating mRNA into protein

12


aa-tRNA competition

I aa-tRNAs compete for matching with codon

I aa-tRNAs classified ascognate, near-cognate, or non-cognate

I fidilty of translation

• insertion errorwrong amino acid added to nascent chain

• insertion timetime for successful elongation, amount of delay

12


peptidyl elongation I

12


peptidyl elongation II

12


probabilistic automaton

0.23 / 80167 / 46

60 / FAST

260 / 0.40190

85

1

0

2 3

5

4 6 8arrival

FAST

FAST

cognate and near-cognate rates differ

12


probabilistic automaton

0.23 / 80167 / 46

60 / FAST

260 / 0.40190

85

1

0

2 3

5

4 6 8arrival

FAST

FAST

cognate and near-cognate rates differ

arrival rate depends on concentration

12


Prism representation

// initial binding[ ] (s=1) -> k1f * c_cogn : (s’=2) & (cogn’=true) ;[ ] (s=1) -> k1f * c_near : (s’=2) & (near’=true) ;[ ] (s=1) -> k1f * c_nonc : (s’=2) & (nonc’=true) ;

// dissociation[ ] (s=2) & ( cogn | near ) -> k2b :

(s’=0) & (cogn’=false) & (near’=false) ;[ ] (s=2) & nonc -> k2bx : (s’=1) & (nonc’=false) ;

// codon recognition[ ] (s=2) & ( cogn | near ) -> k2f : (s’=3) ;[ ] (s=3) & cogn -> k3bc : (s’=2) ;[ ] (s=3) & near -> k3bn : (s’=2) ;

12


Prism representation (cont.)

// rejection[ ] (s=4) & cogn -> k4rc : (s’=5) & (cogn’=false) ;[ ] (s=4) & near -> k4rn : (s’=5) & (near’=false) ;

// accommodation, peptidyl transfer[ ] (s=4) & cogn -> k4fc : (s’=6) ;[ ] (s=4) & near -> k4fn : (s’=6) ;

// no entrance, re-entrance at state 1[ ] (s=0) -> FAST : (s’=1) ;// rejection, re-entrance at state 1[ ] (s=5) -> FAST : (s’=1) ;// elongation[ ] (s=8) -> FAST : (s’=8) ;

12


matching for E. coliEscherichia coli has for

I codon AAU coding amino acid Asparagine has

• 1 cognate anticodons GUU• 1 near-cognate anticodon UUU• 44 non-cognate anticodons

I codon CUC coding amino acid Leucine has

• 1 cognate anticodon GAG• 4 near-cognate anticodons GUG, CAG, UAG, GGG• 41 non-cognate anticodons

I codon AUC coding amino acid Isoleucine has

• 1 cognate anticodon GAG• 8 near-cognate anticodons GUU, CAU, . . .• 37 non-cognate anticodons

12


insertion error

conditional cognate probability for success

P=?[ (s!=0 & s!=5)U(s=8) {(s=2)&cogn} ]

conditional near-cognate probability for success

P=?[ (s!=0 & s!=5)U(s=8) {(s=2)&near} ]

0.23 / 80167 / 46

60 / FAST

260 / 0.40190

85

1

0

2 3

5

4 6 8arrival

FAST

FAST

success for a cognate pcs = 0.508

success for a near-cognate pns = 0.484 · 10−4

12


insertion error (cont.)

cognate probability for success: proper elongation

P=?[ true U((s=8)&cogn) ]

near-cognate probability for success: insertion error

P=?[ true U((s=8)&near) ]

0.23 / 80167 / 46

60 / FAST

260 / 0.40190

85

1

0

2 3

5

4 6 8arrival

FAST

FAST

error probability depends on concentrations

12



strong correlation insertion errors and ratio near-cognates/cognates

12




probability for erroneous insertion

ratio n

ear−

cognate

/ c

ognate

12




P(error) =pn

s · (fn/tot)

pcs · (fc/tot) + pn

s · (fn/tot)

≈pn

s · fn

pcs · fc

∼fn

fc

12


insertion times

average exit times

e.g., dissociation or rejection of near-cognate

R=?[F((s=0)|(s=5)){(s=2)&near}]

0.23 / 80260 / 0.40

1901

0

2 3

5

4 6 8arrival

FAST

FAST

60+67 FAST65 /

success/failure for cognate T cs = 0.03182, T c

f = 9.342 · 10−3

success/failure for near-cognate T ns = 3.251, T n

f = 0.3914

12


insertion times (cont.)

strong correlation insertion times and ratio near-cognates/cognates

12




average insertion time

ratio (

pseudo−

cognate

+ n

ear−

cognate

) / cognate

12




Tins =∑

∞

i=0 (pcf )

ipcs · ((delay for i + 1 cognates) + T c

s )

=∑

∞

i=0 (pcf )

ipcs ·

(i · (T c

f +fn

fcT n

f +fx

fcT x

f ) + T cs

)≈

fn

fcpc

s Tnf∑

∞

i=0 i (pcf )

i

∼fn

fc

12


Conclusion

I Stochastic automata fit nicely• Prism and CSL adequate

I Model of translation confirms experimentsI Piecewise analysis to support explanation

• Better insight compared to simulation

prism - win.tue.nlevink/education/2if35/pdf/2if35-05a.pdf12 / department of mathematics and computer...

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