Probing Inter-Cell Variability using Bulk Measurements

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Probing Inter-Cell Variability using Bulk Measurements Harrison Steel, and Antonis Papachristodoulou ACS Synth. Biol., Just Accepted Manuscript • DOI: 10.1021/acssynbio.8b00014 • Publication Date (Web): 25 May 2018 Downloaded from http://pubs.acs.org on May 27, 2018

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ACS Synthetic Biology

Probing Inter-Cell Variability using Bulk Measurements

Harrison Steel∗ and Antonis Papachristodoulou∗ Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ

E-mail: [email protected]; [email protected]

Abstract The measurement of noise is critical when assessing the design and function of synthetic biological systems. Cell-to-cell variability can be quantied experimentally using single-cell measurement techniques such as ow cytometry and uorescent microscopy. However, these approaches are costly and impractical for high-throughput parallelised experiments, which are frequently conducted using plate-reader devices. In this paper we describe reporter systems that allow estimation of the cell-to-cell variability in a biological system's output using only measurements of a cell culture's bulk properties. We analyse one potential implementation of such a system which is based upon a uorescent protein FRET reporter pair, nding that with typical parameters from the literature it is able to reliably estimate variability. We also briey describe an alternate implementation based upon an activating sRNA circuit. The feasible region of parameter values for which the reporter system can function is assessed, and the dependence of its performance on both extrinsic and intrinsic noise is investigated. Experimental realisation of these constructs can yield novel reporter systems that allow measurement of a synthetic gene circuit's output, as well as the intra-population variability of this output, at little added cost.

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ACS Synthetic Biology 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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A major challenge in the design of reliable synthetic biological systems is assessing and regulating the impact of variability and noise upon their behaviour.

1

In a particular system,

noise sources can be broadly classied as arising from either intrinsic (due to the stochastic nature of the system's constituent biochemical processes) or extrinsic (due to uctuations in the concentration and behaviour of cellular components with which it interacts) sources.

2,3

Extrinsic noise that impacts the behaviour of synthetic circuits can be caused by both shortand long-term phenomena: It can be introduced by temporary uctuations in cellular machinery abundance

4

of synthetic circuits,

(such as as ribosome sequestration, which impacts the translation rate

5,6

7

or uneven distribution of proteins during cell division ), as well as

epigenetic dierences in proteome and cell state that are passed on as cells divide.

810

Total

gene expression noise is thus a function of both cell-wide uctuations, as well as variability in gene-specic regulation.

11

In many cases noise can negatively inuence the function of synthetic biological circuits.

12,13

Fluctuations in the concentration of individual elements of a synthetic network

can propagate throughout the system, impacting the behaviour of other components.

4,14

This has motivated the design and implementation of synthetic systems that reduce variability,

15

for example via inclusion of feedback control architectures.

1618

At the same time,

in certain circumstances noise can be benecial: In natural systems variability between cells provides an evolutionary advantage in changing environments, tion transfer in genetic networks.

21

19,20

and can enhance informa-

Similarly, synthetic systems can be designed that benet

from the stochasticity of gene expression, accurate modelling of their behaviour.

22

making this factor essential in some cases for

23,24

When synthetic biological designs are realised experimentally, assessment of their noise properties and performance variability is therefore a critical part of their characterisation. Fluorescent reporter proteins are frequently used as an output for such systems due to the

2


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ease with which they are measured.

25

Experiments are often repeated many times (under

theoretically identical conditions) using plate-reader hardware

26

to observe variation in their

behaviour, which is then quantied in terms of the variability in a mean experimental outcome (for example, a cell culture's bulk uorescence output). However, measurement of only a cell culture's mean uorescence can disguise important but often unnoticed behaviours, such as bi-modal responses.

28

27

To accurately characterise many systems, measurement of

intra-population cell-to-cell variability is thus required,

27

for which techniques such as uo-

rescence microscopy and ow cytometry (which can measure individual uorescence levels of a large number of cells) are employed extensively.

29

Recently, improvements in the capabil-

ities of these single-cell measurement technologies have facilitated high-throughput studies of cell-to-cell variability.

30

However, single-cell measurement techniques remain time- and

equipment-intensive, making their use unfavourable if large numbers of samples must be measured at regular time intervals, as is the case when parallelised experiments are used to test synthetic circuits over a range of input parameter combinations (often done using a plate reader).

This experimental challenge motivates the aim of this paper:

To design a synthetic

biological reporter circuit that allows characterisation of cell-to-cell variability using only measurements of a cell culture's mean behaviour. Such a reporter could be used in any synthetic biological system with a uorescent protein output to give an estimate of cell-to-cell variability at little added cost.

This would be of great utility for many high-throughput

noise-measurement experiments (which currently require ow cytometry) as it would allow them to be performed eciently and in parallel using a plate-reader device. We achieve noise estimation via a biological implementation of a multiplication function that can be used to measure a variable's mean value, as well as its mean squared value, at a population level. It is then possible to estimate the cell-to-cell standard deviation of a system's output, which previously required measurements at a single-cell level. Thus, though our system does not

3



Page 4 of 34

return the complete shape of the expression distribution (as does ow cytometry), for cases in which the distribution mean and width are the primary parameters of interest (as in many past studies

13,15,17,18

) it can provide an ecient experimental alternative.

We begin the paper with a description of the simple statistics behind this estimation procedure, showing how measurement of only bulk culture parameters can allow quantication of cell-to-cell variability. In the case of log-normally distributed behaviour (as is often found in biological systems

31

), we demonstrate that this provides a direct estimate of the

width of the system's output distribution.

We then describe a uorescent-protein FRET

implementation for such a circuit (an alternate implementation using an activating sRNA circuit is described in Supplementary Section 2), and discuss expected parameter values for its operation. The FRET system's variance-estimating performance is then analysed, and its operational parameter range and sensitivity to both intrinsic and extrinsic noise is assessed. These results are discussed with reference to potential experimental conditions in which our reporter system might be employed, and potential sources for error (as well as alternate implementations which might minimise these) are outlined.

Noise Distributions and Quantication We consider a measurable random biological variable of interest,

X,

which might (for exam-

ple) correspond to the uorescence output from a single cell due to production of a uorescent reporter protein within that cell. The expectation of number of cells. In this context

X , E[X],

is its mean value over a large

E[X] would (approximately) correspond to the total uores-

cence of a cell colony divided by the number of cells of which it is comprised, hence giving the average uorescence per cell.

Many factors are not considered in this approximation,

such as the attenuation of emitted uorescence as it passes through experimental media (thereby making cells further from a detector device appear slightly dimmer), however these

4


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secondary eects will be ignored in our current analysis. The variance of a random variable

X

is given by:

32

V ar[X] = E[X 2 ] − E[X]2 , where

(1)

E[X 2 ] is the expected (arithmetic mean) value of the variable X

to as the second moment of the random variable

X.

squared, also referred

In the current context,

X2

would cor-

respond to the squared uorescence value of a single cell.

Values of

X

for individual cells will vary within a population, and when a large number

of cells are measured individually (i.e. at a single-cell level) the approximate distribution of

X

can be ascertained. For many variables of interest in biological systems, such as the level

of mRNA or protein produced, this distribution is observed to be approximately log-normal during steady growth

33

cells in stationary phase

(though it may be better described by a normal distribution for

34

). It has been shown that a log-normal distribution can emerge in

such systems due to the inherent complexity of the noisy biochemical processes involved.

2 A log-normal distribution for a given variable will be denoted Lognormal(µ,σ ), where

σ

µ and

are the mean and standard deviation respectively of the variable's natural logarithm. The

nth

moment of a log-normal distribution can be calculated analytically:

1

2 σ2

E[X n ] = enµ+ 2 n

where of

31

n

35

,

(2)

is a positive integer. In reality we will not be able to measure the absolute value

E[X 2 ],

but rather a value calculated by our reporter system that is proportional to its

value. We denote the measured second moment

Em [X 2 ] = γm E[X 2 ],

where

γm

reects this

proportionality and is termed the 2nd moment gain.

Experimentally, variation between cells in the value of a parameter in terms of the width of the distribution when the variable

5


X

X

is often quantied

is plot on a log scale (i.e. the


parameter

σ ).

Page 6 of 34

We can derive an expression for this variable for the log-normal distribution

in terms of measurable quantities, giving:

s Em [X 2 ] σ = ln − ln(γm ), Em [X]2 where

σ

on

Em [X]

Em [X 2 ]

changes in

is the measured rst moment (the mean). The square-root-log dependence of is worth noting, as it means small changes in

Em [X 2 ].

σ

will be measurable as large

Even if we do not estimate/measure the parameter

an increasing function of quantity

(3)

Em [X 2 ]/Em [X]2

Em [X 2 ]/Em [X]2 grows, so does

γm , we note that σ

is

(when the argument is real), meaning that as the

σ.

If the underlying distribution is not log-normal

and/or unknown we can estimate the population standard deviation using (1), though again estimation of

γm

is necessary.

Noise strength (fano factor) is a common parameter of interest when characterising the variability of biological systems, and can be used to discriminate between dierent mechanisms contributing to noise in protein abundance.

36,37

This parameter, dened as the ratio of

the expression variance to its mean is constant for a perfectly poissonian process.

38

However,

for most biological process (for which a log-normal distribution is anticipated) the coecient of variation provides a better measure of gene-expression noise.

37

The (arithmethic) Co-

ecient of Variation (CV ) is dened as the ratio of the arithmetic standard-deviation to the mean (SD[X]/E[X] where between these parameters.

39

SD[X] =

p V ar[X])

due to the linear relationship observed

We therefore have:

CV =

p

eσ2 − 1 ≈ σ

where the last approximation uses the truncated Taylor-series small sigma (σ

. 0.8) .

(4)

ex ≈ 1 + x

which is valid for

This highlights the appropriateness of the coecient of variation for

assessing noise in biological systems exhibiting log-normal behaviour: It is proportional to

6


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the width of the expression distribution when plot on a log scale.

We have thus outlined a method for determining the level of variability in a parameter

X,

as is often measured via ow cytometry, but here only using measurements of bulk

parameters. To build an estimator of

Em [X 2 ]

we must create a system with fundamental

structure:

X

where we can both measure

X,

−− +X) −* −Y

and the variable

Y ∝ X 2.

We now describe and analyse a

possible FRET implementation for such a variance estimating circuit. A second potential implementation, based upon an activating sRNA circuit, is described in Supplementary Section 2.

A Biological Implementation: FRET Reporter Pair One approach to measuring a variable (Em [X

2

])

X

in terms of its mean (Em [X]) and second moment

is to use a dimerising FP-FRET (Fluorescent Protein - Fluorescence Resonance

Energy Transfer) protein pair that can approximate a multiplication function (Fig. 1a). In this case we have chosen the mClover3 (C ) and mRuby3 (R) uorescent proteins, other possibilities exist. promoter

Px ,

41

40

but many

Both uorescent fusion proteins are expressed in tandem from the

for which we aim to measure the variability in transcription. This variability

may arise from any upstream processes (e.g. gene regulatory networks) interacting with

Px .

Figure 1

When spatially separated, the uorescent proteins have their usual excitation/emission behaviour. However, when they are brought into close proximity (∼

10nm

or less

41

) it is

possible to excite the complex at the lower excitation wavelength of mClover3 (termed the

7



a)

Px mClover

mRuby

Δ. λC

Δ. λR

δC

/0

δR KOFF

KOFF

mClover

/0

mRuby

(C)

(R)

KON δF

/0

mClover mRuby

(F) 1

b)

0.8

Fss/Css

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 34

10-

0x =1 N

KO

0.6

K ON

0.4

Figure 1:

a)

x10-4

KON =2.5

0.2 0

=6

4

x10-4

KON =1 x10-4 KON = 0.2 x10-4

0

0.5

1

Δ

1.5

A FRET Reporter system for analysis of gene expression variability.

Structure of the system, in which the concentration of a FRET-active dimer (F ) is

approximately proportional to the square of the concentration of its unbound components

C

and

R. C

and

R

each consist of a fusion of a uorescent protein (mClover3 and mRuby3

respectively) with a protein binding domain (brown).

Dashed arrows represent reactions

and their directions (with rate parameters as dened in the text), and degradation/dilution reactions (δ ) represent removal of species to the null state

∅.

The DNA operon (top) consists

of a promoter (Px ), Ribosome Binding Sites (RBS, dark-green), genes (labelled), and a transcriptional terminator. (7)) on the value of

∆.

FSS /CSS (as dened by (6) and KON in units of s−1 . For an ideal

The dependence of the ratio

Colours denote dierent values of

multiplication operation values of

b)

FSS /CSS

would be linear in

∆,

which is best satised for small

KON .

donor ), and measure a change in uorescence in the emission spectrum of mRuby3 (the acceptor ). This occurs via non-radiative energy transfer through long-range dipole to dipole

interactions when the urophores are proximal. FRET exist,

41

42

A diversity of approaches to measuring

though we will concentrate on ratiometric measurements of FRET intensity

which may be performed using a plate reader (or uorescence microscope or ow cytometer).

43

8


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To co-localise the two uorescent proteins such that FRET can occur we fuse them to complementary protein binding domains, as has been done in a number of FRET-based protein localisation studies.

44,45

Each uorescent protein can therefore exist in isolation, or

they can dimerise to form a complex

F

(Fig. 1a). The binding strength of the complementary

domains represents the major parameter for tuning in our system, which could initially be investigated experimentally using a pair with ligand-dependent binding. properties will determine the forward and backward rates (KON and

45,46

Their binding

KOF F ) for the reaction:

KON

C +R− )−−* −− F KOF F

If we assume the concentrations of both

C

and

R

variability that we aim to analyse (the parameter

are proportional to the gene expression

∆

introduced at

C

provides multiplicative ability when the concentrations of

and

Px ),

R

then this system

are greater than

F.

Following the terminology of the previous section the concentration (and hence uorescence) of either mClover or mRuby is a proxy for the variable concentration of

F

represents the variable

Y.

X

which we aim to measure, and the

We can model this system using a system of

dierential equations of the form:

where and

F

C, R

C˙ = ∆λC − δC C − KON CR + KOF F F

(5a)

R˙ = ∆λR − δR R − KON CR + KOF F F

(5b)

F˙ = KON CR − KOF F F − δF F

(5c)

are the concentrations of the mClover and mRuby fusion proteins respectively,

is the concentration of the bound FRET complex.

introduced by

Px

which we aim to measure,

λC,R

is the random scaling factor

are lumped transcription/translation rates

(we model these in a single step for simplicity), and corresponding species.

∆

δC,R,F

are degradation rates for the

By setting the time derivatives to zero it is possible to solve the

system of equations (5) analytically to nd the steady-state (ss) concentrations of each

9



Page 10 of 34

complex, giving:

Css , Rss =

∆λC,R − δF Fss . δC,R

(6)

and

Fss = a(b + ∆c − (∆2 d + 2bc∆ + b2 )0.5 )

(7)

where:

a =1/(2KON δF2 )

(8a)

b =δC δR (KOF F + δF )

(8b)

c =KON δF (λC + λR )

(8c)

2 d =KON δF2 (λC − λR )2

(8d)

For an ideal multiplication operation to occur, we aim to tune system parameters such that

2 2 Fss ∝ ∆2 , which would then mean (approximately) that Fss ∝ Css , Rss .

Observing that

2 ∂Fss ∆=0 ) and rst derivative ( ∂∆ |∆=0 ) are zero, to achieve Fss ∝ ∆

the constant term (Fss |

we

thus desire:

∂2 Fss ab2 (c2 − d) = ∼ Constant, ∂∆2 (∆2 d + 2bc∆ + b2 )1.5 for all

∆.

This criteria is best satised if

(9)

b is large and c and d are small.

large by setting the protein degradation terms to be fast (i.e.

δC,R,F

Here

b can be made

are large). However,

tuning these parameters can be challenging (and may introduce noise): A more eective approach is to focus on of

d

setting

c

and

λC = λR ). KON

d

which can be made small by reducing

KON

is the rate at which the fusion proteins

C

(and in the case and

R

dimerise,

which can be reduced by making the binding of their attached protein domains weak. By minimising

Fss

this constraint also achieves

C, R ∝ ∆

analysis of the opposite scenario, in which a large form, and thus

Fss

Fss

in (6). This follows from an intuitive

means most protein is in its dimerised

will be proportional to whichever of

10

Css


or

Rss

is smaller.

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These results can be rearranged to give:

Fss =

which if

λR = λC

and

δR = δC

∆(λR − λC ) + δC Css KON Css · KOF F + δF δR

(10)

(i.e. both fusion proteins are expressed and degraded at equal

rates), simplies to:

Fss = giving a straight-forward estimate of

KOF F

we have

γm ≈ 1/Kd ,

KON 2 C 2 ≈ γm Css KOF F + δF ss γm

for our simulated system. In the case where

Kd = KOF F /KON

where

(11)

δF

Probing Inter-Cell Variability using Bulk Measurements

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