Robustness of Protein Circuits

Robustness of Protein Circuits

03/15/2010

The robustness of biological systems The robustness principle: the essential functions of a biological

system is nearly independent of biochemical parameters that tend to vary from cell to cell, and change under different conditions.

Robustness of a property of a biological system is a relative concept, it is always referred to with respect to some parameters.

Properties that are not robust with respect to some parameters are called fine-tuned.

Robustness is an important design principle of biological systems seen at different levels:1. Molecular interaction networks with respect to variation of concentrations

of molecules, and the noisy environments;

2. Patterning of tissues as an embryo develops into a fetus with respect to environmental fluctuations;

3. Functions of organ systems with respect to damages and environmental fluctuations.

Bacterial chemotaxis Bacteria can sense some specific chemicals, and move either

against or down the gradients of these chemicals. This process is called bacterial chemotaxis.

Repellents: chemicals that drive bacteria away from them.

Attractants: chemicals that attract bacteria to move towards them.

Bacteria compute temporal gradient Bacteria can detect very small change in the concentration

gradient of an attractant or a repellent: one molecule per cell volume (per micron), in background concentration ranging over five orders of magnitude.

Clearly, bacteria are too small to sense the gradient along the length of its body as do some mammalian cells.

It turns out that bacteria use temporal gradient to guide their motion: they use a biased-random walk strategy to sample space and convert a spatial gradient to a temporal one.

To do so, they compare the current chemical concentration to that in the past.

Runs and tumbles There are two types of random walks for bacterial motion:

During the runs, a bacterial cell reduces its tumbling frequency and thus increases its run time.

1. Runs: motion with almost the same direction, last about 1 sec.

2. Tumbles: motion with random directions, last about 0.1 sec.

Bacterial chemotaxis is achieved through altering tumbling frequency

When moving against the gradient of an attractant, they sense a net increase in concentration. They achieve so by reducing their tumbling frequency, and thus tend to move against the gradient.

When moving along the gradient of a repellent, they sense a net decrease in concentration. This is accomplished by increasing their tumbling frequency, and thus changing their direction randomly.

Therefore, the bacteria achieve chemotaxis through sensing the temporal derivative of the concentration of attractants or repellents: dX/dt.

Runs and tumbles are generated by two types of rotation of the flagella motor.

Two types of rotation of the flagella motor

The bacteria flagella motor [source: Berg HC, Ann. Rev. Biochem 2003]

Each cell has several flagella motors that can rotate either clockwise (CW) or counterclockwise (CCW).

When the flagella motors rotate CCW, the flagella rotate together in a bundle and push the cell forward;

When one of the motor rotates CW, its flagellum breaks from the bundle and causes the cell to tumble about and randomly change its direction.

Bacterial chemotaxis Two types of rotation of flagella motors, CW and CCW result in

two types of motion, runs and tumbles, respectively.

Motors turn CCW

Run TumbleMotor turns CW

When tethered to a surfacethe entire cell rotates, andIndividual motors show two-state behavior

time

CCW

CW

10 sec

Adaptation of sensation Adaptation of sensation: many biological sensory systems tend

to reduce their response intensity to persistent stimuli.

Exact adaptation: the steady-state tumbling-frequency does not depend on the level of attractant.

Tumblingfrequency

1/sec

0 5 10 15 20 250

0.5

1

1.5

Attractantadded

exact adaptation

Time [min]

The chemotaxis signal transduction network of E. coli

Chemicals (ligand) are sensed by receptors on the cell membrane.

There are 5 types receptors in E. coli, each can sense a specific set of chemicals.

Receptors are associated with kinase CheA through an adaptor protein CheW.

W

A

Y P YATPADP

Motor

Increased CW rotation and tumbles

Z

Binding of a repellent to its receptor activates CheA, which phosphorylates CheY.

Phosphorylated CheY-p binds to the motor proteins, and increases the probability of CW rotation, and tumbles.

Repellent

The chemotaxis signal transduction network of E. coli

Binding of an attractant to its receptor deceases the activity of kinase CheA, thus few CheY is phosphorylated.

With few motor proteins bund by phosphorylated CheY-P, the probability of CW rotation of motors decreases, thus the frequency of run increases.

Decreased CW rotation, increased runs

W

A

Y P YATPADP

MotorZ

Attractant

Adaptation of chemotaxis in E. coli

Adaptation of chemotaxis is achieved through a feedback loop made of the methylase CheR and the demethylase CheB.

CheR adds methyl groups to the receptor (methylation). Methylated form of the receptor tends to have higher activity than non-mythylated form to activate CheA.

CheB removes methyl groups from the receptor (demethylation).

CheA also phosphorylates CheB, and activates it.

X

Fine-tuned model for adaptation of chemotaxis We can develop a fine-tuned model to explain the exact

adaptation of chemotaxis in E. coli. Assumptions:

X Xm

1. The receptor, CheX and the adaptor CheW form functional unit X;2. CheR work at zero-order kinetics (ie, it is saturated);3. Methylated Xm has higher activity than non-methylated X.

CheR, VR

CheR, VR

,VB

,V’B

a0

a1

Fine-tuned model for adaptation of chemotaxis Thus, it is the concentration of Xm that determines the activity of

CheY, and thus the probability of tumbling.

According to the assumptions, when no chemicals bund to X, we have,

.

,0

therefore,

,0 state,steady At

,

RVBVRkVX

XkBXVRV

dtdX

XkBXVRV

dtdX

RB

Rm

m

mBR

m

m

mBR

m

• VR: the rate of methylation of X by

CheR

• R: the concentration of R

• VB: the rate of demethylation of X by CheB

• B: the concentration of CheB

Fine-tuned model for adaptation of chemotaxis Assume that the activity of Xm is a0 per receptor (or per mole),

then the total steady state activity of phosphorylation is, .00 mXaA

When a saturating concentration of an attractant is added to the cells, the activity of Xm decreases from a0,to a1(a0 >> a1). Thus, the total activity of phosphorylation at short times after the attractant is added becomes,

.0011 mm XaAXaA

However, a decrease in activity of CheA (part of Xm) leads to reduced activity of CheB (V’B), resulting in an increase in the concentration of Xm (Xm’), and thus an increase in total activity of phosphorylation. At steady state,

. and , '''

12 RVBVRkVXXaA

RB

Rmm

Fine-tuned model for adaptation of chemotaxis Under exact adaptation,

0 1 2 3 4 5 6 7 8 9 100

5

10

15

mXaA 00

mXaA 11

'12 mXaA

Activity

Time

.20 AA

Fine-tuned model for adaptation of chemotaxis Therefore,

.

,

,

,

'1

0

'10

'10

'10

RVBVRVBV

aa

RVBVa

RVBVa

RVBVRkVa

RVBVRkVa

XaXa

RB

RB

RBRB

RB

R

RB

R

mm

If binding of an attractant reduce the activity per concentration by 10 fold, ie, a0/a1=10, and assume VRR=1, VBB=2, and a0=10, then,

.10121100

0

RVBV

kVaARB

R

Fine-tuned model for adaptation of chemotaxis For exact adaptation, ie, A0=A2, we have V’BB=1.1, and a1=1. Therefore, .10

11.111

'1

2

RVBVkVaA

RB

R

Thus, this model depends strictly on the match of parameters to achieve exact adaptation. It is not robust with respect to the changes in these parameters.

To see this, let’s assume CheR is reduced by 20%, so that VRR is now 0.8. thus,

,66.6

8.028.0100

0

RVBV

kVaARB

R

and,

.33.28.01.1

8.01'

12

RVBV

kVaARB

R

Fine-tuned model for adaptation of chemotaxis Thus, a modest 20% of change in CheR level results in almost

a 3-fold change in the activity of phosphorylation of CheY, completely abolishing the exact adaptation.

0 1 2 3 4 5 6 7 8 9 100

5

10

15

Activity

Time

Adaptation is not exact

mXaA 00

'12 mXaA

Barkai-Leibler robust mechanism for exact adaptation of chemotaxis

A mechanism of robustness of exact adaptation of chemotaxis was first suggested by Barkai and Leibler in 1997.

In a simplified Barkai-Leibler model, we assume that:

X

X*m

Xm

2. Binding of an attractant to Xm switches it to inactive state;

3. CheR works at zero order kinetics;

4. CheB only demethylates the active form of X, Xm*.

1. There is a single methylation site on X, non-methylated X (X) is inactive, and methylated X (Xm) can switch between active and inactive states;

VR

VB


The activity of phosphorylation of CheY is proportional to X*m,

at the steady state, dXm/dt=0, the rate of methylation equals that of demethylation,

.)(

,

*

**

*

m

mBR

mm

m

XkBXVRV

dtXXd

aXA

.

,

,

*

*

*

*

RVBVRakVaXA

RVBVRkVX

XkBXVRV

RB

Rm

RB

Rm

m

mBR


Addition of an attractant to the cells, rapidly inactivates Xm*, resulting in a rapid drop of the phosphorylation of CheY, and thus a reduced tumbling frequency.

After this initial drop of activity, adaptation takes place because the substrate of CheB (Xm*) decreases, and Xm continues to increase, and so does Xm* .

0 1 2 3 4 5 6 7 8 9 100

5

10

15

Act

ivity

time

*0 maXA

*2 maXA

X

X*m

XmVR

VB


The activity of CheY is also described by,

At the steady state, dXm/dt=0, the rate of methylation equals that of demethylation,

.)( and , *

***

m

mBR

mmm Xk

BXVRVdt

XXdaXA

This activity is the same as before when no attractant is present. Thus it does not depend on the parameters. This model explains the robustness of the exact adaptation of chemotaxis.

.

, ,

*2

**

*

RVBVRakVaXA

RVBVRkVX

XkBXVRV

RB

Rm

RB

Rm

m

mBR


Variation of parameters (k, VR, VB, R and B), can change the steady state level and other parameters, however, exact adaptation is always maintained.

0 1 2 3 4 5 6 7 8 9 100

5

10

15

0 1 2 3 4 5 6 7 8 9 100

5

10

15

Model parameters:k=10, VR R =1, VB B=2.

A A

time timeSame parameters with R reduced by 20%, results in a decreased steady state level, but does not affect exact adaptation.

1

11, RVBV

RakVARB

Rst

2

22, RVBV

RakVARB

Rst


There are limits to this robustness:

1. When VRR > VBB, the saturation assumption of CheR no longer holds, and the robustness breaks down.

2. We assume CheB only works on Xm*, in reality, CheB may have small effect on Xm, this may result in a loss of exact adaptation by a factor .

Experimental test of robustness in bacterial chemotaxis

Fold-expression is the ratio of CheR protein level to that in the wild-type.

Adaptation precision is the ratio of tumbling frequency before and after saturating attractant.

The exact adaptation is robust with respect to the level of CheR.

The steady stead activity and response time are fine-tune properties and they are inversely correlated.

adaptation time

steady-statetumbling freq.

Alon, Barkai, Surette, Leibler, Nature 1999

The robustness in bacterial chemotaxis can be well explained by the Barkai-Leibler model

The variability of chemotaxis among individuals in terms of different tumbling rate, response time can be explained by the cell-cell variation in chemotaxis protein levels.

The inverse correlation relationship between response time and steady state tumbling frequency can also be derived from the Barkai-Leibler model with multiple methylation sites.

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Robustness of Protein Circuits