systems biology i: bistability macromolecules jonathan weissman

Systems Biology I: Bistability

Macromolecules

Jonathan Weissman

Systems Biology

How does complex biological behavior emerge from the organization of proteins into pathways and networks?

Common properties:1. Bistability

• convert continuous stimulus into discrete response2. Hysteresis

• “memory” of stimulus long after it is withdrawn

Outline for this lecture:

Use simple graphical method to describe conditions thatlead to bistability

Discuss two biological examples of bistability:Xenopus oocyte maturation - MAPK Lac operon

Go through experiments that demonstrate system is bistableUnderstand what the origin of bistability isDescribe the physiological consequences of bistability

Cellular responses to stimuli: switches versus dimmers

Ak1S

k−1I⏐ →⏐ ⏐← ⏐⏐ ⏐ A*

forward rate=k1S A[ ] =k1S Atot[ ] −k1S A*[ ]Atot =A+A*

reverse rate=k−1I A*[ ]

At steady state:

k −1I A*[ ] =k1S Atot[ ] −k1S A*[ ]

A ⏐ →⏐← ⏐⏐ A*

stimulus

inactivation

Michaelian system - no feedback

k1S k −1I

System is monostablePerturb [A*/Atot], return to ss

Assume S, I far fromsaturation

Michaelian system - no feedback, cont.

At steady state:

k −1I A*[ ] =k1S Atot[ ] −k1S A*[ ]A*[ ]Atot[ ]

=S

k−1I / k1+S, when S=k−1I / k1,

A*[ ]Atot[ ]

=0.5

A*[ ]Atot[ ]

=S

EC50 +S, EC50 =S req to produce 50% response

increase S

Response levels off becausethe larger S gets, the less inactive A there is for the kinase to act on, and the moreA* there is for the p’tase

Ak1S

k−1I⏐ →⏐ ⏐← ⏐⏐ ⏐ A*

Michaelian system with linear feedback

forward rate =basal rate + feedback rate

forward rate=k1S A[ ] + k2 A*[ ] A[ ]forward rate=(k1S+ k2 A*[ ])( Atot[ ] − A*[ ])

Extremes:With no [A*] feedback will be 0

When [A*]/[Atot]=1 and [A]=0 sofeedback also equals 0

In between:Will be maximum when [A*]=[A]

feedback rate = k2 A*[ ] A[ ]

e.g. A* increases activity of Sor phosphorylates A in trans


forward rate=k1S A[ ] + k2 A*[ ] A[ ]

Consider simple case wherek1S is small so basal rateis negligible

Michaelian system with linear feedback, cont.

Michaelian system with linear feedback, cont.

strong feedback

weak feedback

k2 Atot[ ] > k−1I

k2 Atot[ ] < k−1I

stable ON state butunstable OFF state

stable OFF statebut no ON state

Two steady states:

One steady state:

How can we stabilize the OFF state?

1. Make feedback sigmoidal function of A*

• three steady-states, one unstable (threshold) and two stable

• at threshold, if decrease A*, back reaction is faster than forward reaction and system is driven to OFF state

• at threshold, if increase A*, forward reaction is faster than back reaction and system is driven to ON state

Making feedback sigmoidal function of A*, cont.

i. cooperativity

feedback ∝ A*[ ]n

ii. zero-order ultrasensitivity

Zero-order conditionsrate independent of [substrate]

rate ∝[substrate]First-order conditions

e.g. A* activates S and takes n molecules of A* to activate S


ii. zero-order ultrasensitivity, cont.

Opposing reactions with the [substrate] >> Km for the modifying enzymes

First-order

Zero-order

Atot =0.01×Km

Atot =100×Km

First-order: changing p’tase by 1.5x has smalleffect on steady state

Zero-order: changing p’taseby 1.5x changes ss from >90%A* to >90% A

A A*kinase

phosphatase


ii. zero-order ultrasensitivity, cont.

• small changes in enzyme activity lead to large changes in modified protein, steady state

• low abundance, high affinity inhibitor of S

• once inhibitor is saturated, S will be effective at causing increases in [A*] (same as S-P below)

iii. inhibitor ultrasensitivity

- inhibitor: 1/9 EC50 causes 10% response90% response at 9 x EC50

+ inhibitor: 10/9 units kinase causes 10% response 90% response at 10 units kinase

81-fold change in kinase 10 to 90%

9-fold change in kinase 10 to 90%

2. Saturate back reaction with A* before feedback saturates


• back reaction saturates as A* accumulates

• back reaction can more than keep up with first bit of feedback, but then is overwhelmed as feedback continues to rise


2. Saturate back reaction with A* before feedback saturates

1. Make feedback a sigmoidal function of A*

i. cooperativity

ii. zero-order ultrasensitivity

iii. inhibitor ultrasensitivity

Systems are bistable only for a limited range of kinetic parameters

Switching between the OFF and ON states

• continuously increase feedback-independent stimulus S

• bistable system converts continuous change in S into discontinuous change in output (A*)


basal rate=k1S A[ ]

increase S - increase slope of basal rate

Switching between the OFF and ON states

What happens when we lower the stimulus?

OFF state reappears when S is lowered to 3, but no driving force to leaveON state

HYSTERESIS/IRREVERSIBILITY

Hysteresis and irreversibility

Hysteresis

Any bistable system will exhibit some degree of hysteresis

Path from ON to OFF is different than that from OFF to ON

Irreversibility

System stays in ON state indefinitely after S is removed

Occurs when feedback is strong

Decreases chattering between ON and OFF states when S is near threshold

Mechanism for biochemical memory - unless positive feedback is broken system will remain in ON state - can remember a stimulus long after itis removed

Example 1: Xenopus oocyte maturation

immature

G2 arrested

mature

metaphase arrest,meiosis II

progesterone

MAPK cascade

How does this system turn a continuously varying stimulus (progesterone) into an all-or-none response?

meiosis IGVBD

Xenopus oocyte maturation, cont.

Measure response of oocytes to different [progesterone] by measuringMAPK phosphorylation (activation)

Response of population - treat oocytes with give [progesterone], makeextract, measure MAPK phosphorylation and tabulate % activation

Response isgraded - Michaelian!


How do we interpret these observations?

[progesterone] [progesterone]

OR

Answer: Look at the response of individuals


Response is all-or-none - no intermediate levels of MAPK activationBISTABLE

Where does this behavior come from?

Clue: Microinjection of activated Mos gives same result (bistability),indicating it comes from pathway downstream of Mos (MAPK cascade)


What are the functional consequences of bistability for this system?

Small change in stimulus near threshold throws switch

At low stimulus buffered a bit from change

No intermediate states


Can quantitate the sensitivity of the response - for a given change in stimulus, how much change In output (MAPK phosphorylation) results

Michaelian - requires 81-fold change in [ligand] to drive system from 10% to 90% on

Ultrasensitive - requires <81-fold change in [ligand] to drive system from 10% to 90% on

Subsensitive - requires >81-fold change in [ligand] to drive system from 10% to 90% on


Hill coefficient - measure of sensitivity

Michaelian: nH=1

Ultrasensitive: nH>1

Subsensitive: nH<1

Relationship exists between Hillcoefficient and frequency of occurrence of intermediate states

Higher Hill coefficient - fewerintermediate states

Counted 190 oocytes - lowerbound on nH is 42


What is the origin of the bistability?

1. Multistep ultrasensitivity - stabilizing the off state

MAPKKMAPKKKà Üà à à àá àà à à à MAPKK-P

MAPKKKà Üà à à àá àà à à à MAPKK-PP

MAPKMAPKKà Üà à à àá àà à à à MAPK-P

MAPKKà Üà à à àá àà à à à MAPK-PP

Two phosphorylation events required for MAPK and MAPKK activation

Activating kinase must dissociate in between phosphorylationevents (to load ATP)

Rate will vary as [upstream kinase]2

Two steps - effects are multiplicative

REMEMBER: Bistability arises from positive feedback anda mechanism to stabilize the off state


1. Multistep ultrasensitivity - stabilizing the off state

• forward rate (slope) varies as [kinase]2

• 1 unit of kinase gives 50% S-P; double kinase and rate increases 4-fold

Michaelian Multistep ultrasensitivity

• S-P must increase to higher level than in Michaelian case for rates to balance and reach new steady state (80% vs. 67%)

• leads to sigmoidal stimulus-response curve

1. Multistep ultrasensitivity, cont. - stabilizing the off state


Measure dependence of MAPK activation on Mos input in vitro in oocyteextracts

nH ~ 5

~2.5 fold change in MAPKKKchanges MAPK from 10%to 90% activity

2. Zero-order ultrasensitivity - stabilizing the off state


• no clear data that reactions in MAPK cascade are operating in zero-order conditions, but concentrations and kinetic properties that have been measured are consistent with this idea

3. Positive feedback

Mos ⏐ →⏐ Mos* ⏐ →⏐ MEK ⏐ →⏐ MAPK

+

• protein synthesis-dependent feedback from active MAPK to Mos

• small change in activity leads to large change in ss position


Is the biochemistry underlying oocyte maturation irreversible?

• Is the [progesterone] required to activate MAPK and Cdc2 different from the [progesterone] required to maintain activities?

Induction: incubate withprogesterone, wait untilmaturation plateaus

Maintenance: incubate with600 nM progesterone, waituntil GVBD plateaus, washfor 10 hr, incubate withdifferent [progesterone]

Maturation is irreversible - “memory” of progesterone


• based on what we learned about bistability, postulate it is the strength of the feedback that gives rise to irreversibility

• predict that if feedback is disrupted, system should lose irreversibility or “memory”

+E2=MAPK pathway activated

+E2, wash=MAPK pathway activated then 16 hr wash

Disrupt feedback withCHX, Mos antisense

“Memory” requires positive feedback

• also predict that disrupting positive feedback should alter bistability - make less ultrasensitive

• quantitate response of individual oocytes to microinjected Mos in presence and absence of CHX

• observe oocytes with intermediate amounts of phosphorylation - loss of bistability



• ultrasensitivity of MAPK cascade plus positive feedback generates bistability

• Michaelian MAPK with feedback - unstable OFF state

• ultrasensitive MAPK with feedback - stable OFF state, filters out small stimuli

Lac operon

• beta-galactosidase is formed by bacterial cells grown in the presence of lactose - beta-gal is necessary for metabolism of lactose

• known that non-metabolizable galactosides (e.g. TMG; thiomethyl-galactoside) induce beta-gal formation

• add high [TMG], see induction of beta-gal at maximal induction rate - under these conditions cells respond uniformly

• at lower [TMG], see dose-dependent rate of induction of beta-gal

• cells that have been exposed to a high [TMG] and then shifted to lower [TMG] (maintenance concentrations) continue to synthesize beta-gal at the maximum rate

Phenotypic change following transient signal -memory of inducer!

Lac operon, cont.

Is the Lac system bistable? Cells growing in [TMG]that gives intermediate levelof beta-gal synthesis (e.g. 30%)

Dilute to maintenance [TMG] togive ~1 bacterium/10 tubes

• 10% of the tubes developed bacterial populations

• 30% of the cultures from single bacteria had maximal beta-gal levels; 70% had only small amounts

Population at intermediate [TMG] consists of cells that are fullyinduced and those that are uninduced - BISTABLE

Lac operon, cont.

LacI = Lac repressor

LacY = Lac permease

TMG

TMG

LacI LacY

LacZ

LacZ = beta-galactosidase

• autocatalytic positive feedback (double negative)

• if there is non-linearity in response, might expect bistability in TMG, LacI activity, LacY and LacZ expression

Lac operon, cont.

net rate =d TMG[ ]

dt=vin −vout

vout =vbind +vcatab =kbind TMG[ ] + 0 =kbind TMG[ ]For vin, know ∝ TMG[ ]eBut TMG also indirectly activates entry of TMG through LacI

Response=TMG[ ]i

K + TMG[ ]in

vin = TMG[ ]e k0 +TMG[ ]i

K + TMG[ ]in

⎛

⎝⎜

⎞

⎠⎟

Need k0 parameter because if there is no permease the firstmolecule of TMG cannot enter the cell, independent of the externalconcentration - either operon is not totally repressed or TMG canslowly diffuse

Could come from TMG bindingLacI, relationship between activeLacI and LacY production…

Lac operon, cont.

[TMG]

[TMG]

Three steady-states:ss1, 3 stabless2 unstable

Bistable

vout =kbind TMG[ ]

vin = TMG[ ]e k0 +TMG[ ]i

K + TMG[ ]in

⎛

⎝⎜

⎞

⎠⎟

Lac operon, cont.

Can see bistability and hysteresis in single cell experiments

Take cells expressing Lac promoter-GFP and grow in different [TMG] - either initially uninduced or induced

Uninduced then grown20 hr in 18 M TMG

Bistability and hysteresis

start uninduced

start induced

[maintenance]

Lac operon, cont.

When inducer is added, permease synthesis will be initiated ata rate determined by inducer in medium

At low [inducer] rate of permease synthesis may be so low that probability of making a permease molecule during lifetime of bacteriumis small

Once bacterium has permease, [inducer] inside cell will increase, whichwill increase probability that second molecule of permease will be formed

Two permeases further increases rate of permease - autocatalytic rise tomaximal permease

Once have maximal permease, bacterium and progeny will be inducedindefinitely since [inducer] inside is above maintenance concentration

Maintenance can be explained if the number of permease molecules is largerelative to threshold, there is high probability that each daughter cell willreceive sufficient permease to insure induction

Transfer to less than maintenance - observe exponential decrease in activity,equivalent to constant chance of becoming uninduced - partitioning betweendaughter cells - some will not get enough to insure induction

Flipping the switch: induction at intermediate lac concentration occurs by complete dissociation of the tetrameric lac repressor

(A) A high concentration of intracellular inducer can force dissociation of the repressor from its operators, (B) At low or intermediate concentrations of intracellular inducer, partial dissociation from one operator by the tetrameric LacI repressor is followed by a fast rebinding. Consequently, no more than one transcript is generated during such a brief dissociation event. However, the tetrameric repressor can dissociate from both operators stochastically and then be sequestered by the inducer so that it cannot rebind, leading to a large burst of expression. (C) A time-lapse sequence captures a phenotype-switching event. In the presence of low inducer, one such cell switches phenotype to express many LacY-YFP molecules (yellow fluorescence overlay) whereas the other daughter cell does not

Bistability Biobliography Reviews & Comments on Research Articles:

Ferrell, J. E., Jr. (1996). Tripping the switch fantastic: how a protein kinase cascade can convert graded inputs into switch-like outputs. Trends Biochem Sci 21, 460-6.

Koshland, D. E., Jr. (1998). The era of pathway quantification. Science 280, 852-3.

Koshland, D. E., Jr., Goldbeter, A., and Stock, J. B. (1982). Amplification and adaptation in regulatory and sensory systems. Science 217, 220-5.

Laurent, M. and Kellershohn, N. (1999) Multistability: a major means of differentiation and evolution in biological systems. Trends Biochem. Sci. 24, 418-422. Research articles:

Novick, A. and Weiner, M. (1957). Enzyme induction as an all-or-none phenomenon. Proc. Natl. Acad. Sci. U S A 43, 553-566.

Cohn, M. and Horibata, K. (1959). Analysis of the differentiation and of the heterogeneity within a population of Escherichia coli undergoing induced beta-galactosidase synthesis. J. Bacteriology 78, 613-633. Ferrell, J. E., Jr., and Machleder, E. M. (1998). The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes [see comments]. Science 280, 895-8.

Goldbeter, A., and Koshland, D. E., Jr. (1981). An amplified sensitivity arising from covalent modification in biological systems. Proc Natl Acad Sci U S A 78, 6840-4.

Goldbeter, A., and Koshland, D. E., Jr. (1984). Ultrasensitivity in biochemical systems controlled by covalent modification. J. Biol. Chem. 259, 14441-14447.

Huang, C. Y., and Ferrell, J. E., Jr. (1996). Ultrasensitivity in the mitogen-activated protein kinase cascade. Proc Natl Acad Sci U S A 93, 10078-83.

Xiong, W. and Ferrell, J. E., Jr. (2003). A positive-feedback-based bistable 'memory module' that governs a cell fate decision. Nature 426, 460-465.

Ozbudak, E. M., Thattai, M., Lim, H. H., Shraiman, B. and van Oudenaarden, A. Multistability in the lactose utilization network of Eschericia coli. Nature 427, 737-740. Tanaka M, Collins S, Toyama B, Weissman JS. (2006). The physical basis of how prion conformations determine strain phenotypes. Nature 442, 585-589.

Choi PJ, Cai l,Freda K, Xie XS (2008) A stochastic single gene event triggers switching of a bacterial celll. Science 332. 442-446.

systems biology i: bistability macromolecules jonathan weissman

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