# March 30

## 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
• Chemical species : Xi
• Either different molecule (repressors one and two)
• Same molecule different conformation (different conformation of DNA)
• Same molecule in a different location
• Reaction routes
• Some reactions are a function of I (flipping a bit)
• Rule : bit flips when reaction takes place
• State
• Given by the array of species
• Can't measure every concentration
• Interrogate a subset of the system
• 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
• 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
• In the time interval, Tu, there is a probability of changing from Xo to Xi
• P(Tc>t) = e^(-Iu*t)
• Requirment : Tu>Tc
• 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
• 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*)
• 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!)

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 ...
• 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

• Flippee
• System

• 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

**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