Introduction to Biophysics - bgoodlab · 2021. 8. 2. · Introduction to Biophysics 1010 101 1 1010...

APPHYS 205 // BIO 126/226:Introduction to Biophysics

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Biology generates exquisitely complex behavior from simple chemical building blocks

e.g. a human white blood cell “chasing” an S. aureus bacterium (credit: David Rogers)


e.g. firing pattern of a single “grid neuron” in a rat moving within a box (credit: Derdikman, Whitlock, Waade, Moser & Moser


…but physics imposes strong constraints!


…but physics imposes strong constraints!

Can be different from our “usual” physical intuition:

Physics of everyday objects Quantum mechanics

Biology generates exquisitely complex behavior …but physics imposes strong constraints

Physics of everyday objects Quantum mechanicsBiophysics

interesting new physics emerges at these cellular scales



Major themes:

• Thermal fluctuations (statistical mechanics)

• Constant jostling of particles in solution (diffusion)

• Finite # molecules per cell (counting noise)



Major themes:

• Quantitative reasoning & order-of-magnitude estimation

Phillips, Kondev, Theriot, & Garcia

“biology by the numbers”

tour of basic length, time, & number scales in biology

Today:

E. Coli will be our standard ruler

0 1um 2um

1um

2um

0Farka et al (2014)

(10 m)-6

Volume: ~10 L-15

Mass: ~10 g-12

(assuming density ~ water)

Taking the molecular census of an E. Coli cell

DNA mRNA Protein (amino acids) Growthtranscription translation

(RNA polymerase) (ribosome)

~1-2(5x10 bp)6

Taking the molecular census of an E. Coli cell

DNA mRNA Protein (amino acids) Growthtranscription translation

(RNA polymerase) (ribosome)

~1-2(5x10 bp)6

1 protein ~300 amino acids ~10 total aa9

Human cells have a huge diversity of structure and function

Budding yeast

E. coli

Red blood cell

Fibroblast

Neuron~1mm-1m

~5um

Biological processes occur over a huge range of timescales

~20-60min


~25 aa/s

~40 bp/s

What can we learn from these numbers?

~25 aa/s


~25 aa/s

+

~300 aa / protein ~10 aa / cell9


~25 aa/s

+


Minimum replication

time= ~10 aa9


~25 aa/s

+


Minimum replication

time= ~10 aa9 ~25 aa / s/ x ~2 x10 ribosomes4


~25 aa/s

+


Minimum replication

time= ~10 aa9 ~25 aa / s/ x ~2 x10 ribosomes4 = ~30 mins!


~25 aa/s

+


Minimum replication

time= ~10 aa9 ~25 aa / s/ x ~2 x10 ribosomes4 = ~30 mins!

~10s per protein(~20s / mRNA)

How can we count absolute # of molecules per cell?

One way: fluctuations during division

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Each molecule flips a coin:

“binomial distribution”

Gene Regulation at theSingle-Cell Level

Nitzan Rosenfeld,1* Jonathan W. Young,3 Uri Alon,1

Peter S. Swain,2 * Michael B. Elowitz3 .

The quantitative relation between transcription factor concentrations and therate of protein production from downstream genes is central to the functionof genetic networks. Here we show that this relation, which we call the generegulation function (GRF), fluctuates dynamically in individual living cells,thereby limiting the accuracy with which transcriptional genetic circuits cantransfer signals. Using fluorescent reporter genes and fusion proteins, wecharacterized the bacteriophage lambda promoter PR in Escherichia coli. Anovel technique based on binomial errors in protein partitioning enabledcalibration of in vivo biochemical parameters in molecular units. We foundthat protein production rates fluctuate over a time scale of about one cellcycle, while intrinsic noise decays rapidly. Thus, biochemical parameters,noise, and slowly varying cellular states together determine the effectivesingle-cell GRF. These results can form a basis for quantitative modeling ofnatural gene circuits and for design of synthetic ones.

The operation of transcriptional genetic cir-cuits (1–5) is based on the control of pro-moters by transcription factors. The GRF isthe relation between the concentration ofactive transcription factors in a cell and the

rate at which their downstream gene productsare produced (expressed) through transcrip-tion and translation. The GRF is typicallyrepresented as a continuous graph, with theactive transcription factor concentration onthe x axis and the rate of production of itstarget gene on the y axis (Fig. 1A). The shapeof this function, e.g., the characteristic level ofrepressor that induces a given response, andthe sharpness, or nonlinearity, of this response(1) determine key features of cellular behaviorsuch as lysogeny switching (2), developmen-tal cell-fate decisions (6), and oscillation (7).Its properties are also crucial for the designof synthetic genetic networks (7–11). Cur-rent models estimate GRFs from in vitro

data (12, 13). However, biochemical parame-ters are generally unknown in vivo and coulddepend on the environment (12) or cell history(14, 15). Moreover, gene regulation may varyfrom cell to cell or over time. Three funda-mental aspects of the GRF specify the behav-ior of transcriptional circuits at the single-celllevel: its mean shape (averaged over manycells), the typical deviation from this mean,and the time scale over which such fluctua-tions persist. Although fast fluctuations shouldaverage out quickly, slow ones may introduceerrors in the operation of genetic circuits andmay pose a fundamental limit on their ac-curacy. In order to address all three aspects, itis necessary to observe gene regulation in in-dividual cells over time.

Therefore, we built Bl-cascade[ strains ofEscherichia coli, containing the l repressorand a downstream gene, such that both theamount of the repressor protein and the rateof expression of its target gene could bemonitored simultaneously in individual cells(Fig. 1B). These strains incorporate a yellowfluorescent repressor fusion protein (cI-yfp)and a chromosomally integrated target pro-moter (PR) controlling cyan fluorescent pro-tein (cfp). In order to systematically varyrepressor concentration over its functionalrange (in logarithmic steps), we devised aBregulator dilution[ method. Repressor pro-duction is switched off in a growing cell, sothat its concentration subsequently decreasesby dilution as the cell divides and grows intoa microcolony (Fig. 1C). We used fluores-cence time-lapse microscopy (Fig. 1D; fig.S1 and movies S1 and S2) and computationalimage analysis to reconstruct the lineage tree(family tree) of descent and sibling relationsamong the cells in each microcolony (fig.

1Departments of Molecular Cell Biology and Physicsof Complex Systems, Weizmann Institute of Science,Rehovot, 76100, Israel. 2Centre for Non-linear Dy-namics, Department of Physiology, McGill University,3655 Promenade Sir William Osler, Montreal, Quebec,Canada, H3G 1Y6. 3Division of Biology and Depart-ment of Applied Physics, Caltech, Pasadena, CA 91125,USA.

*These authors contributed equally to this work.To whom correspondence should be addressed.E-mail: [email protected]

Fig. 1. Measuring agene regulation func-tion (GRF) in individualE. coli cell lineages. (A)The GRF is the depen-dence of the produc-tion rate of a targetpromoter ( y axis) onthe concentration ofone (or more) tran-scription factors (x ax-is). (B) In the l-cascadestrains (16) of E. coli,CI-YFP is expressedfrom a tetracyclinepromoter in a TetRþbackground and canbe induced by anhydro-tetracycline (aTc). CI-YFP represses produc-tion of CFP from the PRpromoter. (C) The reg-ulator dilution experi-ment (schematic): Cells are transiently induced to express CI-YFP and thenobserved in time-lapse microscopy as repressor dilutes out during cell growth(red line). When CI-YFP levels decrease sufficiently, expression of the cfp targetgene begins (green line). (D) Snapshots of a typical regulator dilution

experiment using the OR2*–l-cascade strain (see fig. S3) (16). CI-YFP proteinis shown in red and CFP is shown in green. Times, in minutes, are indicated onsnapshots. (Insets) Selected cell lineage (outlined in white). Greater timeresolution is provided in fig. S1.

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25 MARCH 2005 VOL 307 SCIENCE www.sciencemag.org1962

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Rosenfeld et al, Science 2005



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S2). For each cell lineage, we quantified overtime the level of repressor (x axis of theGRF) and the total amount of CFP protein(Fig. 2A). From the change in CFP over time,we calculated its rate of production (y axis ofthe GRF) (16).

Regulator dilution also provides a naturalin vivo calibration of individual protein fluo-rescence. Using the lineage tree and fluores-cence data, we analyzed sister cell pairs justafter division (Fig. 2B). The partitioning ofCI-YFP fluorescence to daughter cells obeyeda binomial distribution, consistent with anequal probability of having each fluorescentprotein molecule go to either daughter (16).Consequently, the root-mean-square error inCI-YFP partitioning between daughtersincreases as the square root of their total CI-YFP fluorescence. Using a one-parameter fit,we estimated the fluorescence signal ofindividual CI-YFP molecules (Fig. 2B andsupporting online material). Thus, despitecellular autofluorescence that prohibits detec-tion of individual CI-YFP molecules, obser-vation of partitioning errors still permitscalibration in terms of apparent numbers ofmolecules per cell.

The mean GRFs obtained by these tech-niques are shown in Fig. 3A for the PR pro-moter and a point mutant variant (fig. S3).These are the mean functions, obtained byaveraging individual data points (Fig. 3B) inbins of similar repressor concentration, indi-cating the average protein production rate at agiven repressor concentration. Their coopera-tive nature would have been Bsmeared out[ bypopulation averages (6, 17, 18).

These mean GRF data provide in vivovalues of the biochemical parameters under-lying transcriptional regulation. Hill func-tions of the form f(R) 0 b/E1 þ (R/kd)

n^ areoften used to represent unknown regulation

functions (1, 6–10). Here, kd is the con-centration of repressor yielding half-maximalexpression, n indicates the degree of effectivecooperativity in repression, and b is the maxi-mal production rate. Hill functions indeed fitthe data well (Fig. 3A and Table 1). The mea-sured in vivo kd is comparable to previousestimates (2, 12, 13, 19) (see supporting onlinetext). The significant cooperativity observed(n 9 1) may result from dimerization of repres-sor molecules and cooperative interactionsbetween repressors bound at neighboring sites(2, 12, 13, 19, 20). A point mutation in theOR2 operator, OR2* (20) (fig. S3), significant-ly reduced n and increased kd (Fig. 3A andTable 1). Note that with similar methods it iseven possible to measure effective coopera-tivity (n) for native repressors without fluores-cent protein fusions (16).

We next addressed deviations from themean GRF. At a given repressor concentra-tion, the standard deviation of productionrates is È55% of the mean GRF value. Suchvariation may arise from microenvironmen-tal differences (21), cell cycle–dependentchanges in gene copy number, and varioussources of noise in gene expression and othercellular processes (22). We compared micro-colonies in which induction occurs at differ-ent cell densities (16). The results suggestedthat the measured GRF is robust to possibledifferences among the growth environmentsin our experiments (fig. S6). We analyzedthe effect of gene copy number, which variestwofold over the cell cycle as DNA repli-cates. The CFP production rate correlatedstrongly with cell-cycle phase; cells about todivide produced on average twice as muchprotein per unit of time as newly dividedcells (16). Thus, gene dosage is not com-pensated. Nevertheless, after normalizing pro-duction rates to the average cell-cycle phase

(16), substantial variation still remains in theproduction rates, and their standard devia-tion is È40% of the mean GRF (Fig. 3). Thedeviations from the mean GRF show a log-normal distribution (see supporting online textand fig. S5).

These remaining fluctuations may arisefrom processes intrinsic or extrinsic to geneexpression. Intrinsic noise results from sto-chasticity in the biochemical reactions at anindividual gene and would cause identicalcopies of a gene to express at different levels.It can be measured by comparing expressionof two identically regulated fluorescent pro-teins (22). Extrinsic noise is the additionalvariation originating from fluctuations incellular components such as metabolites,ribosomes, and polymerases and has a globaleffect (22, 23). Extrinsic noise is often thedominant source of variation in E. coli andSaccharomyces cerevisiae (22, 24).

To test whether fluctuations were of intrin-sic or extrinsic origin, we used a Bsymmetricbranch[ strain (16) that produced CFP andYFP from an identical pair of PR promoters(Fig. 4D, movie S3). The difference betweenCFP and YFP production rates in these cellsindicates È20% intrinsic noise in proteinproduction Eaveraged over 8- to 9-min in-tervals (16)^, suggesting that the extrinsiccomponent of noise is dominant and con-tributes a variation in protein productionrates of È35%.

Our measurements provide more detailedanalysis of extrinsic noise in two ways. First,in previous work (22), extrinsic noise includedfluctuations in upstream cellular components,including both gene-specific and global fac-tors. Here, we quantify the extrinsic noise atknown repressor concentration, and so extrin-sic noise encompasses fluctuations in globalcellular components such as polymerases or

Fig. 2. Data and calibration.(A) Fluorescence intensitiesof individual cells are plottedover time for the experimentof Fig. 1D. Red indicates CI-YFP, which is plotted on a log-arithmic y axis to highlight itsexponential dilution: As CI-YFPis not produced, each divisionevent causes a reduction ofabout twofold in total CI-YFPfluorescence. Green indicatesCFP, which is plotted on alinear y axis to emphasize itsincreasing slope, showing thatCFP production rate increasesas the CI-YFP levels decrease.A selected cell lineage is high-lighted (also outlined in Fig.1D). (B) Analysis of binomial errors in protein partitioning to find vy, theapparent fluorescence intensity of one independently segregating fluores-cent particle (16). Cells containing Ntot copies of a fluorescent particle(total fluorescence Ytot 0 vy I Ntot) undergo division (inset). If eachparticle segregates independently, N1 and N2, the number of copiesreceived by the two daughter cells, are distributed binomially, and satisfy

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r

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=2. A single-parameter fit thus determines the value of

vy. Here we plot kN1 – N2k=2 (in numbers of apparent molecule dimers) versusNtot 0 N1þ N2. Blue dots show the scatter of individual division events. Crosses(red) show the root-mean-square (RMS) error in protein partitioning and itsstandard error. The expected binomial standard deviation is shown in black.

R E P O R T S

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