Bioinformatics: Network Analysis Kinetics of Regulatory Networks: Basic Building Blocks COMP 572 (BIOS 572 / BIOE 564) - Fall 2013 Luay Nakhleh, Rice University 1
Basic Building Blocks ✤ Here we show how simple signaling pathways can be embedded in networks using positive and negative feedback to generate more complex behaviors - toggle switches and oscillators - which are the basic building blocks of the dynamic behavior shown by non-linear control systems. 2
Protein Synthesis and Degradation: Linear Response Using the law of mass action, we have: S dR dt ¼ k 0 þ k 1 S � k 2 R ; R S: signal strength (concentration of mRNA) R: response magnitude (concentration of protein) 3
Protein Synthesis and Degradation: Linear Response Using the law of mass action, we have: S dR dt ¼ k 0 þ k 1 S � k 2 R ; R Steady-state solution: R ss ¼ k 0 þ k 1 S k 2 4
Phosphorylation and Dephosphorylation: Hyperbolic Response dt ¼ þ � S Using the law of mass action, we have: ATP ADP dR P dt ¼ k 1 S ð R T � R P Þ � k 2 R P : R RP H 2 O P i R P : concentration of the phosphorylated form of the response element R T : total concentration of the response element 5
Phosphorylation and Dephosphorylation: Hyperbolic Response dt ¼ þ � S Using the law of mass action, we have: ATP ADP dR P dt ¼ k 1 S ð R T � R P Þ � k 2 R P : R RP Steady-state solution: H 2 O P i R T S R P ; ss ¼ ð k 2 = k 1 Þ þ S : 6
Linear and Hyperpolic Responses ✤ Linear and hyperbolic curves share the properties of being graded and reversible: ✤ Graded means that the response increases continuously with signal strength. A slightly stronger signal gives a slightly stronger response. ✤ Reversible means that if the signal strength is changed from S initial to S final , the response at S final is the same whether the signal is being increased (S initial >S final ) or decreased (S initial >S final ). 7
Phosphorylation and Dephosphorylation: Buzzer Assuming Michaelis-Menten kinetics: S ATP ADP dt ¼ k 1 S ð R T � R P Þ dR P k 2 R P � K m 1 þ R T � R P k m 2 þ R P R RP Steady-state is a solution of the equation: H 2 O P i k 1 S ð R T � R P Þð K m 2 þ R P Þ ¼ k 2 R P ð K m 1 þ R T � R P Þ : The biophysically acceptable solution (0<R P <R T ) of this equation is: R P ; ss ¼ G ð k 1 ; S ; k 2 ; K m 1 ; K m 2 Þ ; R T R T R T where the Goldbeter-Koshland function, G, is defined as: G ð u ; v ; J ; K Þ 2 uK ¼ : q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð v � u þ vJ þ uK Þ 2 � 4 ð v � u Þ uK v � u þ vJ þ uK þ 8
Phosphorylation and Dephosphorylation: Buzzer Sigmoidal 1 Response (R P ) 0.5 0 0 1 2 3 Signal (S) A sigmoidal response is continuous and reversible, but abrupt. The element behaves like a buzzer, where one must push hard enough on the button to activate the response. In terms of phosphorylation, the signal S must be strong enough to create a noticeable change of the equilibrium. 9
Perfect Adaptation: Sniffer dR R ss ¼ k 1 k 4 k 3 dt ¼ k 1 S � k 2 X � R X k 4 k 2 k 3 S dX X ss ¼ k 3 S k 1 dt ¼ k 3 S � k 4 X k 4 R k 2 feed-forward The response mechanism exhibits perfect adaptation to the signal. 5 Although the signaling pathway responds transiently to changes Adapted 1.9 4 in signal strength, its steady-state response R SS is independent of S S R and is only controlled by the ratio of the four kinetic rates of the 3 X system. 2 1.4 Such behavior is typical of chemotactic systems, which respond to 1 an abrupt change in attractants or repellents, but then adapt to a 0 constant level of the signal. –1 0.9 Our own sense of smell operates this way; hence, this element is 0 10 20 termed a sniffer. Time 10
Positive Feedback: One-way Switch R activates enzyme E (by phosphorylation), and EP S k 1 k 2 enhances the synthesis of R: R k 4 k 3 EP E feedback a Goldbeter-Koshland function In the response curve, the control system is found to be bistable Mutual activation between 0 and S crit . In this regime, there are two stable steady- Response (R) 0.5 state response values (on the upper and lower branches, the solid lines). The is called a one-parameter bifurcation. Which value is S crit taken depends on the history of the system. After the signal threshold S crit has been crossed once, the system will remain on the upper curve. This is termed a one-way switch. Apoptosis is an example for this behavior, where the decision to shut down the 0 0 10 cell must be clearly a one-way switch. Signal (S) 11
Mutual Inhibition: Toggle Switch The same as the previous case, with the only difference S that E now stimulates degradation of R. R dR dt ¼ k o þ k 1 S � k 2 R � k 0 2 E ð R Þ � R EP E E ð R Þ ¼ G ð k 3 ; k 4 R ; J 3 ; J 4 Þ 1 This systems leads to a discontinuous behavior. This type of Mutual inhibition bifurcation is called a toggle-switch. If S is decreased enough after Response (R) starting from a high level, the switch will go back to the off-state on the lower curve meaning a small response R. For intermediate S crit1 0.5 stimulus strength (S crit1 <S<S crit2 ), the response of the system can be S crit2 either small or large, depending on the history of S(t). This is often called hysteresis. Biological examples of such behavior include the lac operon in bacteria. 0 0 1 2 Signal (S) 12
Negative Feedback: Homeostasis The response element, R, inhibits the enzyme E S k 1 catalyzing its synthesis. R k 2 k 4 k 3 EP E 1 Homeostatic Response (R) This type of regulation is called homeostasis. It is sort of an adaptation, but not a sniffer, because stepwise increases in S do 0.5 not generate transient changes in R. 0 0 1 2 Signal (S) 13
Negative Feedback: Oscillation S Second scenario: k 1 k 2 X k 3 k 7 (2) Y YP k 4 k 5 RP R k 6 (1) Two possible ways of inhibition 0.5 1 Steady-state is unstable YP S crit1 S crit2 X 5 Response (R P ) 0.4 between S crit1 and S crit2 ; it 0.3 0.5 oscillates between R Pmin 0.2 0.1 and R Pmax . RP 0 0.0 0 0 2 4 6 0 25 50 Signal (S) Time 14
Putting It All Together: Cell Cycle Control System 15
(a) M / G 1 m o d u l e Damaged Cdc25 DNA P APC APC Cdk1 P CycB Cdc20 Cdc25 P Unreplicated e l DNA u Wee1 d Misaligned o m Cdc20 chromosomes M / 2 P APC G Wee1 P Cdk1 CycB Cdk1 CKI CKI Cdk1 CycB Growth CycB G 1 / S m o d P u CKI CKI l e CKI CKI P P Wiring diagram for the cyclin-dependent kinase (Cdk) network regulating DNA synthesis and mitosis. 16
(a) M / G 1 m o d u l e Damaged Cdc25 DNA P APC APC Cdk1 P CycB Cdc20 Cdc25 P Unreplicated e l DNA u Wee1 d Misaligned o m Cdc20 chromosomes M / 2 P APC G Wee1 P Cdk1 CycB Cdk1 CKI CKI Cdk1 CycB Growth CycB G 1 / S m o d P u CKI CKI l e CKI CKI P P toggle-switch (mutual inhibition between Cdk1- cyclin B and CKI) 17
toggle-switch (mutual activation between Cdk1-cyclin B and Cdc25, and mutual inhibition between Cdk1-cyclin B and Wee1) (a) M / G 1 m o d u l e Damaged Cdc25 DNA P APC APC Cdk1 P CycB Cdc20 Cdc25 P Unreplicated e l DNA u Wee1 d Misaligned o m Cdc20 chromosomes M / 2 P APC G Wee1 P Cdk1 CycB Cdk1 CKI CKI Cdk1 CycB Growth CycB G 1 / S m o d P u CKI CKI l e CKI CKI P P 18
oscillator, based on negative-feedback loop. Cdk1-cyclin B activates the APC, which activates Cdc20, which degrades cyclin B. (a) M / G 1 m o d u l e Damaged Cdc25 DNA P APC APC Cdk1 P CycB Cdc20 Cdc25 P Unreplicated e l DNA u Wee1 d Misaligned o m Cdc20 chromosomes M / 2 P APC G Wee1 P Cdk1 CycB Cdk1 CKI CKI Cdk1 CycB Growth CycB G 1 / S m o d P u CKI CKI l e CKI CKI P P 19
Acknowledgment ✤ Material is based on the paper ✤ “ Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell ”, Tyson et al. , Current Opinion in Cell Biology, 15: 221-231, 2003. 20
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