# March 30

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## Chemical counter

- System

- Input : what we want to count
- Variable I that is continuous with respect to time, given by I=f(t)
- Examples: iptg, light, chemical species
- Count peak of I up to some time, T

- Peak : I(t)'=0, I(t)
*>0 (maximum), above threshold, and minimum between peaks lower than another threshold*

- Input : what we want to count

- Chemical species : Xi
- Either different molecule (repressors one and two)
- Same molecule different conformation (different conformation of DNA)
- Same molecule in a different location

- Chemical species : Xi

- Reaction routes
- Some reactions are a function of I (flipping a bit)
- Rule : bit flips when reaction takes place

- Reaction routes

- State
- Given by the array of species
- Can't measure every concentration
- Interrogate a subset of the system

- State

- Predict
- N(t)
- X(t) represents the state of X at time T, which could be Xi, Xii, etc.
- This is a good counter : P(X(t))=X(5)=1, if t=5

- Predict

- Example
- X0 at time=0
- Assume discrete input with T(up) and T(low)
- Rules:
- Do nothing when signal is low
- When high: xi->x(i+1) with rate r and time Tci
- Time given by Tci = time of reaction = exponential distributed with respect to rate = Iu
- Probability that Tci > T; Iu*r^(-Iu*t)
- Probability that Tci > t is given by e^(-Iu*t)
- The chance does not decrease with respect to time

- Example

- In the time interval, Tu, there is a probability of changing from Xo to Xi
- P(Tc>t) = e^(-Iu*t)
- Requirment : Tu>Tc

- In the time interval, Tu, there is a probability of changing from Xo to Xi

- What can go wrong:
- Remain at Xo, thus Tc<Tu, with Tc driven by reaction rate given by PDF: P(tc > t)=e^(-Iu*t)
- Key points
- Iu needs be fast enough to allow Tc<Tu
- Not too fast such that Tci+Tc2<Tu

- What can go wrong:

- P(correct) at first count; Tci<Tu and Tci+Tcii>Tu
- Tci=T*, want T*<Tu
- Tcii+T*>Tu; Tcii>Tu-T*
- Integral ( P(Tci=T*) P(Tcs>Tu-T*) dT
- P(Tci=T*) from PDF = Iu*-IuT*
- P(Tcs>Tu-T*) from PDF = Iu*-Iu(Tu-T*)

- P(correct) at first count; Tci<Tu and Tci+Tcii>Tu

- Tl = nothing happens and Tu=pulse; only change state during Tu and Tl can be long, short, unequal, etc

- xi->x(i+1) with rate proportional to Iu given by PDF: Iu*e^-(Iu*t)

- P(X(k*Tu))=Xk=Probability of correct counting at count value K

- Key rules
- tci+tcii ...tck < kTu ; exactly k events over kTu
- tci+ ...tc(k+1) > kTu
- Possion process: probability of k events occur over timescale T = e^(-lamba*T)*(lambda*T)^k.(k!)
- Interval between each event
- Lambda is the rate at which each event occurs: Iu, same as Iu in tu ~ Iu*e^(-Iu*t)
- Time is k*Tu

- Probability that one counts correctly: how will this change if I change Iu, Tu,

- Input signal changes over time
- It peaks up at some point
- During time interval, Tu, there is probability of state change

- Probability of change state tci ~ Iu*e^(-Iu*t)
- Driven species : Iu
- More molecules

- Successful count given by
- tci < Tu
- tcii+tcII>Tu

- I want to have K state changes over total induction time K*Tu
- Implies that intermediate state changes don't matter
- For example two count within one large induction time and no change within the second

- Probability of this happening is given by Poisson distribution
- P(k, Iu) = e^(-lamba*T)*(lambda*T)^k.(k!)

- Implies that intermediate state changes don't matter

Iu is the reaction rate for flipping driven by:

- Want k spots within kTu

## What data can we get now?

- We can induce and get uni-directional flip
- We can measure growth in fluorescent signal

## What are the critical issues

- How fast does the flippee flip?
- Current data lumps all events, so we need to dis-aggregate

## How do we de-couple the events?

- Big question: what is the rate limiting factor in the process?

- When is time between induction and binding?
- Localization assay with flipper-GFP fusion

- When is time between induction and flipping?
- PCR with respect to time
- Bulk assay, but don't know distribution
- How do we do this quantitatively?
- Now, we amplify spontaneous flipping ...

- PCR with respect to time

- How fast does it take to visualize GFP?
- Measure signal with respect to time following induction under control of same promoter

- What drives the time of flipping?
- Inducer length
- Int / Xis expression dynamics

## Modeling

- Flippee
- System

## General experimental tools

- Reporter
- Gemini

## Measure Int / Xis leakiness

- Put reporter behind the Int or Xis promoter
- Measure signal when not induced

- Tag the Int with fusion
- Direct measurement of the flipper

## mRNA quantification

- mRNA coding for flipper (int)
- Tag mRNA with florescence to quantify

## Measurement of DNA binding

- FP-Int fusion
- Low copy Int expression and visualize signal only when bound

## Recombination

- Bulk measurements for timescale of flipping
- Plate reader

****PCR with respect to time**

- Induce
- Stop reaction
- PCR

- How sensitive is the PCR single with respect to the number of flipped templates?

- Single cell measurements for timescale of flipping
- See variability across the population
- Get the distribution

- FP on plasmid measurement
- Distance between two signal cases is ~100-80nm on plasmid
- Plasmid replicates and is not synchronized with the cell
- Therefore two dots could come from