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Network motifs in the neuronal network of C elegans

• C elegans has around 1000 cells including some 300 neurons – the connections have been mapped

• Network has many of the motifs found in transcriptional and signal transduction networks

• Scale is different (size: cells, time: milliseconds) than transcription networks (size: nano-meter, time: minutes)

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Feed Forward Loop

• Activity of neuron:time-dependent transmembrane voltagedifference

• Classic model based on summation of synaptic inputs (integrate-and-fire)

w1 and w2 are strengths of synaptic connections

�: step functionCan result in AND or OR gates

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Function

• Activity of Z determined by:

• Experiments indicate:Z=(X1 OR X2) AND Y

• Z is activated by either X1 or X2 but onlyif it is persistent (as it takes a bit of time~10 ms to activate Y)

• Can also act as coincidence detector

X1 X2

Y

Z

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Multilayer perceptrons in C elegans

As in signal-transduction networks but more connections between nodesat the same layer. Unknown if they function in same way.

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Chapter 7: Robustness – example:Bacterial chemotaxis

• Movement pattern of E coli adapts to environment (attractants, repellents)

• Movement affected by ”tumbling frequency”

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Motor – direction of rotation

• All motors run counter-clockwise (CCW) – the cell is pushed forward

• One/some motors run clockwise – tumble about, randomization of direction

• Figure 7.4

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Immediate reaction to adding attractant –and exact adaptation

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Chemotaxis transduction network

X

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First signal pathway

• Attractant binds receptor• Binding of attractor lowers activity of receptor (X)

– Receptor activity – proportion of receptors in active state

• Active receptor can cause modification of CheY – a response regulator protein – that diffuses into the cell– CheY is phosporylated (becoming CheY-P)

• CheY-P can bind to flagella motor and increase probability that rotation switches from CCW to CW (and cell tumbling)– CheZ dephosphorylates CheY-P – at steady state level of CheY-P is

(and tumble frequency) constant

• Thus attractant binding receptor decreases tumbling frequency

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Adaptation by second pathway

• Receptor can be methylated (0-4/5 methylations possible per receptor)

• Methylation of receptor done by CheR – demethylation by CheB– Methylated form of receptor is called Xm– Xm more active than X

• Active X (receptor) acts to phosporylate CheB – making it more active

• So reduced X activity means less active CheB – and less demethylation by CheB– X becomes more active

• So X first becomes less active due to attractant binding and then gradually becomes more active due to methylation– Methylation process much slower than main pathway from X to CheY

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How is exact adaption achieved?

• Fine-tuned model– Variation in (some) parameters implies that exact adaptation is not

achieved– Fails to explain robustness observed experimentally

• Robust model– Exact adaptation achieved independently of e.g., (exact) abundance of

regulating proteins– Additional assumptions about system behavior– Robustness verified experimentally

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Fine tuned model

• First natural model• Let Xm be the ratio of methylated X (receptors)• Assume that only Xm is active and has rate a0 per methylated

receptor• CheR is methylating – and works at saturation (independent

of the concentration of its substrate), rate VR. • CheB is demethylating and works with Michaelis-Menton

kinetics• Rate of change of Xm is

where R and B denote concentration of CheR and CheB

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Steady state

• Rate of change of Xm is

where R and B denote concentration of CheR and CheB.• Steady state is at

• Steady-state activity in absence of attractant:

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How exact adaptation is achieved

• Assume that attractant is added so that all receptors bind attractant ligand– Receptors assume inactive state and activity per receptor becomes

– where a1<<a0

• This accounts for sharp initial drop in tumbling frequency• Because the receptors are less active, CheB action

(demethylation) is reduced from VB to VB’• Therefore receptor methylation Xm begins to increase

(continuous methylation by CheR)• Receptor methylation reaches steady state

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New steady state in fine-tuned model

• Activity is then

• In order to have exact adaption, we need to have

• Example shows that 10% change in CheR abundance makes huge impact

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Robust model for exact adaptation

• Unmethylated X – inactive• Methylated Xm – switches between active and inactive state

Xm* and Xm

• Activity is proportional to Xm*

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Key features in robust model

• CheR – methylating – works at saturation• CheB can only demethylate active receptors Xm*• Change in abundance of methylated X is then

where R and B are concentrations of CheR and CheB• Steady state at: Giving activity:

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What happens?

• When attractant is added, probability of active receptor is decreased– Initial sharp decrease in tumbling frequency

• Since CheB only works on active receptors, the number of methylated receptors increases (since CheR continues to methylate):

• Steady state

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How does this work?

• ”The number of active receptors adjusts itself so that demethylation balances the reference flux of methylation”– Demethylation more frequent when activity is high – making for less

activity– Regulation via methylation happens on slower timescale than regulation

due to binding of attractants and repellors

• Robustness – robustness of exact adaptation with respect to variation of parameters– K, VR, VB, R and B.

• Experiments show that exact adaption is achieved even if CheR is changed 100-fold (Alon et al 1999)

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Tumbling frequency as a function of time for wild-type (RP437) cells. Circles: cells stimulated at time t = 0 by mixing with saturating attractant (1 mM L-aspartate). Squares: unstimulated cells (mock-mixed with chemotaxis buffer). Tumbling frequency was determined using computerized video tracking14.

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