Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence Shortening of Telomeres and Replicative Senescence Sarah Eugene joint work with Thibault Bourgeron, Philippe Robert and Zhou Xu UPMC, INRIA and IBPC March 8, 2016
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence O UTLINE Biological Framework and Experiments Telomeres Evolving with Telomerase If telomeres were always repaired More Accurate Model Replicative senescence The Model Time of Senescence
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence D EFINITIONS ◮ Telomere: non-coding sequences at the end of chromosomes ◮ Replicative Senescence: state of a cell unable to divide
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence D EFINITIONS ◮ Telomere: non-coding sequences at the end of chromosomes ◮ Replicative Senescence: state of a cell unable to divide = ⇒ the replication machinery implies a shortening of telomeres = ⇒ when too short, the cell enters in replicative senescence (otherwise loss of genetic information)
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence T ELOMERES ARE FASHIONABLE IN CURRENT BIOLOGY Telomeres are involved in: ◮ Aging
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence T ELOMERES ARE FASHIONABLE IN CURRENT BIOLOGY Telomeres are involved in: ◮ Aging ◮ Cancer
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence S EMI - CONSERVATIVE DNA R EPLICATION
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence S EMI - CONSERVATIVE DNA R EPLICATION Replication Forks
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence S EMI - CONSERVATIVE DNA R EPLICATION Replication Forks
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence T HE T ELOMERE E ND P ROBLEM
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence T HE T ELOMERE E ND P ROBLEM
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence T HE T ELOMERE E ND P ROBLEM 5’ 3’ 3’ 5’ DNA Replication 5’ 3’ 3’ 5’ + 5’ 3’ 3’ 5’
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence M OTIVATIONS ◮ In stem cells and germ cells, telomeres are repaired by a protein, the telomerase
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence M OTIVATIONS ◮ In stem cells and germ cells, telomeres are repaired by a protein, the telomerase ◮ In somatic cells, the telomerase is inhibited: the telomeres are only shortened until they are too small to allow replication
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E XPERIMENTS ◮ haploids lineages in Saccharomyces cerevisiae
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E XPERIMENTS ◮ haploids lineages in Saccharomyces cerevisiae ◮ first: telomeres are repaired by the telomerase ( ↔ beginning of life)
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E XPERIMENTS ◮ haploids lineages in Saccharomyces cerevisiae ◮ first: telomeres are repaired by the telomerase ( ↔ beginning of life) ◮ then: the telomerase is inhibited, the cells enter in replicative senescence ( ↔ aging) http://www.nature.com/ncomms/2015/150709/ncomms8680/extref/ncomms8680-s3.mov
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence Mathematical Goals ◮ Model these two phases (obviously) http://www.nature.com/ncomms/2015/150709/ncomms8680/extref/ncomms8680-s3.mov
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence Mathematical Goals ◮ Model these two phases (obviously) ◮ Describe the equilibrium of the first phase http://www.nature.com/ncomms/2015/150709/ncomms8680/extref/ncomms8680-s3.mov
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence Mathematical Goals ◮ Model these two phases (obviously) ◮ Describe the equilibrium of the first phase ◮ From the time of senescence, estimate the parameters of this equilibrium (’inverse problem’) http://www.nature.com/ncomms/2015/150709/ncomms8680/extref/ncomms8680-s3.mov
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence O UTLINE Biological Framework and Experiments Telomeres Evolving with Telomerase If telomeres were always repaired More Accurate Model Replicative senescence The Model Time of Senescence
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence Q UALITATIVE B EHAVIOUR previous experiments at nucleotide resolution prove that: ◮ the elongation doesn’t depend on telomere length M. Teixeira et al., Telomere length homeostasis is achieved via a switch between telomerase- extendible and -nonextendible states. Cell, 2004.
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence I F TELOMERES WERE ALWAYS REPAIRED ... ◮ L n : length of telomere at n th generation ◮ a : shortening rate ◮ G : geometric random variable of parameter p (elongation) Model L n +1 = ( L n − a ) + + G (1)
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E QUILIBRIUM D ISTRIBUTION ◮ L ∞ equilibrium distribution of ( L n ) n (if exists) ◮ π k = P ( L ∞ = k )
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E QUILIBRIUM D ISTRIBUTION ◮ L ∞ equilibrium distribution of ( L n ) n (if exists) ◮ π k = P ( L ∞ = k ) � u ( L ∞ − a ) + + G � E ( u L ∞ ) = E
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E QUILIBRIUM D ISTRIBUTION ◮ L ∞ equilibrium distribution of ( L n ) n (if exists) ◮ π k = P ( L ∞ = k ) � u ( L ∞ − a ) + + G � E ( u L ∞ ) = E Generating function of L ∞ ( p − 1) u a + p (1 + u + u 2 + ... + u a − 1 ) � � � u L ∞ � E a − 1 � 1 + 1 1 � = pu a � π k u + ... + u a − k k =0
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E QUILIBRIUM : I DENTIFYING ( π 0 , ...π a − 1 ) Normalisation condition a − 1 � p π k ( a − k + 1) = ap − (1 − p ) k =0 Rouch´ e’s Theorem: ( p − 1) u a + p (1 + u + u 2 + ... + u a − 1 ) � � has a − 1 roots in the unit disk iff ap > 1 − p ,
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E QUILIBRIUM : I DENTIFYING ( π 0 , ...π a − 1 ) Normalisation condition a − 1 � p π k ( a − k + 1) = ap − (1 − p ) k =0 Rouch´ e’s Theorem: ( p − 1) u a + p (1 + u + u 2 + ... + u a − 1 ) � � has a − 1 roots in the unit disk iff ap > 1 − p , the ergodic condition.
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence Q UALITATIVE B EHAVIOUR previous experiments at nucleotide resolution prove that: ◮ the elongation doesn’t depend on telomere length ◮ tendency to elongate rather short telomeres M. Teixeira et al., Telomere length homeostasis is achieved via a switch between telomerase- extendible and -nonextendible states. Cell, 2004.
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence M ORE A CCURATE M ODEL ◮ L n : length of telomere at n th generation ◮ a : shortening rate ◮ B : Bernouilli random variable parameter 1 / 2 ◮ G : geometric random variable parameter p (elongation) ◮ i S : elongation threshold Model L n +1 = ( L n − a.B ) + + G ✶ { L n ≤ i s } (2)
✶ Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E QUILIBRIUM ◮ L ∞ equilibrium distribution of ( L n ) n (always exists) ◮ π k = P ( L ∞ = k )
✶ Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E QUILIBRIUM ◮ L ∞ equilibrium distribution of ( L n ) n (always exists) ◮ π k = P ( L ∞ = k ) ◮ a = 1
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E QUILIBRIUM ◮ L ∞ equilibrium distribution of ( L n ) n (always exists) ◮ π k = P ( L ∞ = k ) ◮ a = 1 � � u ( L ∞ − 1) + + G ✶ { Ln ≤ is } E ( u L ∞ ) = E
Biological Framework and Experiments Telomeres Evolving with Telomerase Replicative senescence E QUILIBRIUM ◮ L ∞ equilibrium distribution of ( L n ) n (always exists) ◮ π k = P ( L ∞ = k ) ◮ a = 1 � � u ( L ∞ − 1) + + G ✶ { Ln ≤ is } E ( u L ∞ ) = E Generating function of L ∞ i s E ( u L ∞ ) = (1 − p )(1 + u ) p � u k π k + 1 − u (1 − p ) π 0 (3) 1 − u (1 − p ) k =0
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