physics of bio-molecules and cells

If lac repressor is bound to the operator sequence, thendownstream gene expression is blocked.. Lactose molecules bind reversibly to the repressor protein.. Reactivity: The lac Repressor

Lecturers xi

Course 1 Physics of Protein-DNA Interaction

1.1 The central dogma and bacterial gene expression 3

1.1.1 Two families 3

1.1.2 Prokaryote gene expression 5

1.2 Molecular structure 8

1.2.1 Chemical structure of DNA 8

1.2.2 Physical structure of DNA 10

1.2.3 Chemical structure of proteins 12

1.2.4 Physical structure of proteins 14

2 Thermodynamics and kinetics of repressor-DNA interaction 16 2.1 Thermodynamics and the lac repressor 16

2.1.1 The law of mass action 16

2.1.2 Statistical mechanics and operator occupancy 19

2.1.3 Entropy, enthalpy, and direct read-out 20

2.1.4 The lac repressor complex: A molecular machine 23

2.2 Kinetics of repressor-DNA interaction 26

2.2.1 Reaction kinetics 26

2.2.2 Debye–Smoluchowski theory 28

2.2.3 BWH theory 30

2.2.4 Indirect read-out and induced fit 32

3.3.1 Structural sequence sensitivity 50

3.3.2 Thermal fluctuations 52

4 Electrostatics in water and protein-DNA interaction 53 4.1 Macro-ions and aqueous electrostatics 54

4.2 The primitive model 56

4.2.1 The primitive model: Ion-free 57

4.2.2 The primitive model: DH regime 57

4.3 Manning condensation 58

4.3.1 Charge renormalization 58

4.3.2 Primitive model: Oosawa theory 59

4.3.3 Primitive model: Free energy 61

4.4 Counter-ion release and non-specific protein-DNA interaction 63

4.4.1 Counter-ion release 63

4.4.2 Nucleosome formation and the isoelectric instability 64

Course 2 Mechanics of Motor Proteins

9 Effect of force on the rates of chemical reactions 82

10 Absolute rate theories 85

11 Role of thermal fluctuations in motor reactions 87

Course 3 Modelling Motor Protein Systems

1 Making a move: Principles of energy transduction 98

1.1 Motor proteins and Carnot engines 98

1.2 Simple Brownian ratchet 99

1.3 Polymerization ratchet 100

1.4 Isothermal ratchets 103

1.5 Motor proteins as isothermal ratchets 104

1.6 Design principles for effective motors 105

2 Pulling together: Mechano-chemical model of actomyosin 108 2.1 Swinging lever-arm model 108

2.2 Mechano-chemical coupling 110

2.3 Equivalent isothermal ratchet 111

2.4 Many motors working together 112

2.5 Designed to work 115

2.6 Force-velocity relation 116

2.7 Dynamical instability and biochemical synchronization 118

2.8 Transient response of muscle 119

3 Motors at work: Collective properties of motor proteins 119 3.1 Dynamical instabilities 119

3.2 Bidirectional movement 120

3.3 Critical behaviour 121

3.4 Oscillations 124

3.5 Dynamic buckling instability 125

3.6 Undulation of flagella 127

4 Sense and sensitivity: Mechano-sensation in hearing 129 4.1 System performance 129

4.2 Mechano-sensors: Hair bundles 130

4.3 Active amplification 131

4.4 Self-tuned criticality 133

4.5 Motor-driven oscillations 134

4.6 Channel compliance and relaxation oscillations 136

1.1 Introduction 147

1.1.1 Intrinsic dependence of bond strength on time frame for breakage 148

1.1.2 Biomolecular complexity and role for dynamic force spectroscopy 148

1.1.3 Biochemical and mechanical perspectives of bond strength 150 1.1.4 Relevant scales for length, force, energy, and time 153

1.2 Brownian kinetics in condensed liquids: Old-time physics 154

1.2.1 Two-state transitions in a liquid 155

1.2.2 Kinetics of first-order reactions in solution 156

1.3 Link between force – time – and bond chemistry 158

1.3.1 Dissociation of a simple bond under force 158

1.3.2 Dissociation of a complex bond under force: Stationary rate approximation 159

1.3.3 Evolution of states in complex bonds 163

1.4 Testing bond strength and the method of dynamic force spectroscopy 164

1.4.1 Probe mechanics and bond loading dynamics 165

1.4.2 Stochastic process of bond failure under rising force 168

1.4.3 Distributions of bond lifetime and rupture force 169

1.4.4 Crossover from near equilibrium to far from equilibrium unbonding 172

1.4.5 Effect of soft-polymer linkages on dynamic strengths of bonds 175

1.4.6 Failure of a complex bond and unexpected transitions in strength 177

1.5 Summary 185

Part 2: P Williams and E Evans 186 2 Dynamic force spectroscopy II Multiple bonds 187 2.1 Hidden mechanics in detachment of multiple bonds 187

2.2 Impact of cooperativity 188

2.3 Uncorrelated failure of bonds loaded in series 191

2.3.1 Markov sequence of random failures 191

2.3.2 Multiple-complex bonds 193

2.3.3 Multiple-ideal bonds 194

2.3.4 Equivalent single-bond approximation 195

2.4 Uncorrelated failure of bonds loaded in parallel 198

2.4.1 Markov sequence of random failures 198

2.4.2 Equivalent single-bond approximation 198

2.5 Poisson statistics and bond formation 199

2.6 Summary 203

Seminar 1 Polymerization Forces by M Dogterom 205 Course 5 The Physics of Listeria Propulsion by J Prost 215 1 Introduction 217 2 A genuine gel 218 2.1 A little chemistry 218

2.2 Elastic behaviour 220

3 Hydrodynamics and mechanics 220 3.1 Motion in the laboratory frame 220

3.2 Propulsion and steady velocity regimes 221

3.3 Gel/bacterium friction and saltatory behaviour 223

4 Biomimetic approach 225 4.1 A sphericalListeria 225

4.2 Spherical symmetry 226

4.3 Steady state 227

4.4 Growth with spherical symmetry 229

4.5 Symmetry breaking 229

4.6 Limitations of the approach and possible improvements 231

2 Physics of the actin based cytoskeleton 249

2.1 Actin is a living semiflexible polymer 2492.2 Actin network as viscoelastic body 2532.3 Correlation between macroscopic viscoelasticity and molecular

motional processes 258

3 Heterogeneous actin gels in cells and biological function 260

3.1 Manipulation of actin gels 2603.2 Control of organization and function of actin cortex

by cell signalling 265

5 Activation of endothelial cells: On the possibility

of formation of stress fibers as phase transition of actin-network

7 Controll of cellular protrusions controlled by actin/myosin

Course 7 Cell Adhesion as Wetting Transition?

3 Microinterferometry: A versatile tool to evaluate adhesion

4 Soft shell adhesion is controlled by a double well interfacial

5 How is adhesion controlled by membrane elasticity? 297

6 Measurement of adhesion strength by interferometric contour

7 Switching on specific forces: Adhesion as localized dewetting

8 Measurement of unbinding forces, receptor-ligand leverage

9 An application: Modification of cellular adhesion strength

Course 8 Biological Physics in Silico

1 Introduction: The need to control flows in 2 1/2 D 319

Lecture 1b: Dielectrophoresis and Microfabrication 335

2.1 Fabrication 337

2.2 Viscosity 338

2.3 Electronics and imaging 338

2.4 DNA samples 338

3 Results 339 3.1 Basic results and dielectrophoretic force extraction 339

Lecture 2b: The DNA Prism 366

4 Microfabrication 398

Course 9 Some Physical Problems in Bioinformatics

Course 10 Three Lectures on Biological Networks

1 Enzymatic networks Proofreading knots:

1.1 Length scales and energy scales 439

1.2 DNA topology 440

1.3 Topoisomerases 441

1.4 Knots and supercoils 444

1.5 Topological equilibrium 446

1.6 Can topoisomerases recognize topology? 447

1.7 Proposal: Kinetic proofreading 448

2.4 Some simplifying assumptions 464

2.5 Probeset analysis 466

2.6 Discussion 470

3 Neural and gene expression networks: Song-induced gene expression in the canary brain 471 3.1 The study of songbirds 472

3.2 Canary song 473

3.3 ZENK 474

3.4 The blush 476

3.5 Histological analysis 476

3.6 Naturalvs artificial 479

3.7 The Blush II: gAP 480

3.8 Meditation 481

Course 11 Thinking About the Brain

7 Speculative thoughts about the hard problems 564

Matter has many states, including soft condensed, inert or alive The latter is farfrom thermodynamic equilibrium, and apparently has an agenda of its own Yetthe same physical laws apply to all matter The difference is in the complexity towhich living systems have evolved, to states that gather and process information,replicate themselves, etc.

Molecular and cell biology have dramatically expanded our knowledgeabout this complexity in the last decades This knowledge is the foundation ofbiological physics, which is currently expanding rapidly and is itself adding tothis knowledge Its role in biology is a wonderful challenge: to draw the linebetween necessity and possibility, between results of immutable physical lawsand results of evolution that may be specific to the one natural history we haveaccess to The study of life is, after all, similar to reverse engineering1 Whatfascinating engineering it describes, however! The deeper one gets into thedetails, the more captivating the study becomes: these systems were “designed”bottom-up, so answers to some of the biggest questions about Life are hidden intheir smallest parts

The 75th Les Houches summer school addressed the physics ofbiomolecules and cells In biological systems ranging from single biomolecules

to entire cells and larger biological systems, it focused on aspects that requireconcepts and methods from physics for their analysis and understanding Theschool opened with two parallel lecture series by Robijn Bruinsma and Jonathon

Howard Physics of Protein-DNA Interaction by Robijn Bruinsma started from

the structure of DNA and associated proteins, and lead to discussions ofelectrostatic interactions between proteins and DNA, and the diffusive search for

specific binding sites Joe Howard’s lectures on Mechanics of Motor Proteins

discussed mechanical properties of individual proteins and motors, and ofcomplex cytoskeletal structures Simultaneously, Evan Evans’ shorter series

Using Force to Probe Chemistry of Biomolecular Bonds and Structural Transitions explored the rich dynamic behaviors of rupturing individual

biomolecular bonds These lectures were followed by Erich Sackmann’s

discussion of Micro-rheometry of Actin Networks and Cellular Scaffolds He

gave an introduction to membranes and the cytoskeleton and discussed themechanical properties of cells and the physics of cell adhesion Robijn Bruinsma

1

Reverse engineering: “the process of analysing a subject system to identify thesystem’s components and their interrelationships and create representations ofthe system in another form or at a higher level of abstraction” (E.J Chikofsky

and J.H Cross, II IEEE Software 7 (1990) 13-17.)

examples of information processing in the visual system of the fly, he moved tofundamental questions on how nervous systems process information In

Bioinformatics and Statistical Mechanics Eric Siggia reviewed decoding of

genetic information obtained from genome projects It was followed by Bob

Austin’s lectures on Micro- and Nanotechnology-Physics in Biotechnology, new

technologies which make it possible to study and manipulate biomolecules inartificial arrays and structures Marcelo Magnasco wrapped up the school

excellently with his Three Lectures on Biological Networks, which covered the

unknotting of DNA, the analysis of gene chip data, and studies of geneexpression in learning canaries, all in a style that kept the attention of theaudience to the last minute of four weeks

The lectures were complemented by invited seminars given by Albert

Libchaber (RecA Polymerization on Single-Stranded DNA and Directed Evolution: A Molecular Study) and Marileen Dogterom (Polymerization Forces).

Two public lectures were given in the town of Les Houches, by Albert Libchaber

(Qu'est-ce que la vie ?) and Thomas Duke (Les moteurs de la vie) Furthermore,

Phil Williams gave an invited lecture within Evan Evan's series, and Tom

McLeish and Chris Wiggins contributed with seminars: The Mysterious Case of Too Many β-Sheets and Into Physical Models of Biopolymers, respectively.

During a study period, Tom McLeish gave a well-attended tutorial on thermallyactivated barrier crossing, on the school's lawn, with Mt Blanc as a backdropand most illustrative barrier We also organized sixteen short studentpresentations over four evenings, and two poster sessions with a total ofseventeen posters; see titles and presenters at the end of this volume Thestudents had great energy and enthusiasm, and, amazingly in view of theirschedule, kept it up till the very end

This school had three times as many applicants as there are seats in thelecture hall, and we had to turn down many strong applicants We hope this book

to some extent makes up for this unfortunate restriction on admission

side vis-à-vis Mt Blanc is perfect for learning and interacting As are long hikes

in the mountains on weekends Life-long friendships are formed, we know: Two

of this school’s organizers first met as students in a Les Houches summer school

If the present school has taught and inspired its participants as much as thatschool did years ago, we have done well

costs for some US residents We thank them all for making the school possible

We are convinced that this book presents outstanding examples of biologicalphysics, and thank the contributors again for their great efforts of lecturing andwriting

Henrik FlyvbjergFrank JülicherPál OrmosFrançois David

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and Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, Postbus 9506, 2300 Leiden, The Netherlands

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Contents

1.1 The central dogma and bacterial gene expression 31.2 Molecular structure 8

2 Thermodynamics and kinetics of repressor-DNA interaction 16

2.1 Thermodynamics and the lac repressor 162.2 Kinetics of repressor-DNA interaction 26

3 DNA deformability and protein-DNA interaction 34

3.1 Introduction 343.2 The worm-like chain 403.3 The RST model 50

4 Electrostatics in water and protein-DNA interaction 53

4.1 Macro-ions and aqueous electrostatics 544.2 The primitive model 564.3 Manning condensation 584.4 Counter-ion release and non-specific protein-DNA interaction 63

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PHYSICS OF PROTEIN-DNA INTERACTION

R.F Bruinsma

1 Introduction

1.1 The central dogma and bacterial gene expression

1.1.1 Two familiesLife is based on a symbiotic relationship between two families of biopoly-mers: DNA and RNA, constituted of nucleic-acids, and proteins, consti-tuted of amino-acids [1] Proteins are the active agents of the cell As

enzymes, they control the rates of biochemical reactions taking place inside

the cell They are responsible for the transcription of the genetic code, i.e.,

the production of copies of short segments of the genetic code that are used

as blue-prints for the production of new proteins, and for the duplication

of the genetic code, i.e., the production of a full copy of the genetic code

during cell division Synthesis of other macromolecules, such as lipids andsugars, is carried out by proteins, the mechanical force of our muscles is gen-erated by specialized proteins adept at “mechano-chemistry”, they detectlight, sound, and smell, and maintain the structural integrity of cells

If we view the cell as a miniature chemical factory that simultaneouslyruns many chemical processes, then the proteins form the control system ofthe factory, turning reactions on and off The control system obeys ordersfrom the central office: the cell nucleus The DNA inside the nucleus can

be considered as the memory of the computer system of the central office:

it is the information storage system of the cell Blueprints for the synthesis

of proteins are stored in the form of DNA base-pair sequences, much like

strings of zero’s and one’s store information in digital computers A gene is

the data string required for the production of one protein (actually, multiplevariants of a protein can be produced from the same gene) The beginningand end points of a gene are marked by special “start” and a “stop” signals

When a protein has to be synthesized, a specialized copying protein, RNApolymerase, transcribes a copy of a gene beginning at the start signal andending at the stop signal (see Fig 1)

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EDP Sciences, Springer-Verlag 2002

Fig 1 Gene transcription.

This copy is in the form of an RNA strand known as mRNA (or senger” RNA) A huge molecular machine, the Ribosome, synthesizes the protein from the mRNA blueprint Interestingly, these Ribosomes are com- pound constructs of RNA strands (known as rRNA) and proteins, with the

“mes-active biochemistry carried out not by the protein part, as you might haveexpected, but by the RNA part Indeed, unlike DNA, RNA strands are infact capable to act as enzymes

The information stream is strictly one way: DNA contains the

informa-tion required for the synthesis of proteins The genetic code is not altered

by the transcription, and RNA strands do not insert their code into DNA

We call this basic principle of biochemical information flow the “central

dogma” We know next to nothing about how this elaborate relationshipbetween the nucleic and amino acids developed The basic chemical struc-ture of the two families is quite different The molecular biology of livingorganisms is all highly similar and based on the central dogma and we donot know of the existence of more primitive molecular information and con-trol systems from which we could somehow infer a developmental history(though we suspect that once upon a time both information storage and en-zymatic activity was based purely on RNA since RNA is able to carry out

enzymatic activity as we saw) The central dogma applies to living

organ-isms Retroviruses are able to insert their RNA code into host DNA, using aspecial enzyme called “reverse transcriptase” This looks like an exception

to the central dogma but viruses are not considered living organisms sincethey are not able to reproduce themselves independently nor do they carryout metabolic activity, the two defining requirements of a living organism

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available The motivation behind a study of the interaction between DNAand proteins is quite different from that of a study of synthetic polymers

In polymer physics, we want to compute the free energy and correlation

functions of a typical polymer in a solution or melt, with results that are

as much as possible independent of the detailed molecular structure of thepolymers That philosophy does not apply to biopolymers where we are

dealing with highly a-typical molecules that carry out certain functions.

Their structure presumably evolved under the adaptive pressures exerted

on micro-organisms that relied for their survival on efficient performance ofthe functions these molecules are involved with A molecular biophysicisttries to shed light on how functional molecular devices work and how theirdesign constraints are met These are of course very complex systems, so it

is a good strategy to focus as much as possible on basic principles of physics

of general validity and relying as little as possible on assumptions concerningthe detailed molecular structure The hope is that this will provide us withconstraints on the design and operation of functional biopolymers in the waythat the Second Law of Thermodynamics constrains the maximum efficiency

we will see what insights thermodynamics, statistical mechanics, elasticitytheory, and electrostatics can provide us in this respect

1.1.2 Prokaryote gene expressionHow does an organism “know” when to turn gene transcription on andwhen to turn it off? We divide cells in two groups: eukaryotes and prokary-otes The cells of animals and plants – the eukaryotes – have their DNAsequestered inside a nucleus and the cell has a complex set of internal “or-gans” called organelles Gene expression of eukaryotic cells, the focus ofmuch current research, is a complex affair, which we will discuss in a latersection Bacteria, prokaryotes, lack a nucleus and organelles and their geneexpression is much better understood [3] We will discuss a simple example:

the expression of the “lac” gene of the bacterium Escirichia Coli (E.Coli forshort) [4]

Large numbers of the E.Coli parasitic bacteria live inside your intestines(“colon”) When you drink a glass of milk, part of it will be metabolized

not by you but by your E.Coli bacteria The first step is the breakdown

of lactose, sugar molecules consisting of two linked molecular rings

Lac-tose is broken down into two single-ring glucose molecules This chemical

reaction requires an enzyme, called “β Galactosidase”, to proceed because

lactose does not dissociate spontaneously (an enzyme speeds up a tion by lowering the activation energy barrier) First though, the lactosemolecules must be transferred from the exterior of the bacterium to the cellinterior (or “cytoplasm”) across the membrane that surrounds E.Coli This

reac-is done by another protein, called “Permease” Finally, a third protein,called “Transacetylase”, is required for chemical modification of the sugarmolecules

The DNA of E.Coli carries three separate genes for the production ofthese three enzymes: lacY, lacZ, and lacA Expression of the three genesstarts when the environmental lactose concentration rises, and it stops whenthe lactose concentration drops (to avoid wasteful use of precious macro-molecular material) The three genes are located right behind each other onthe DNA, and – sensibly – they are transcribed collectively Such a cluster

of functionally connected genes is called an “operon” The lac operon also

contains three regulatory sequences:

a) Promoter SequenceThis sequence is “recognized” by RNA Polymerase By that we meanthat RNA Polymerase molecules in solution bind to PromoterSequences on the DNA but not to other sequences From this start

site, RNA polymerase can transcribe RNA in either direction In

one direction, “downstream”, it produces the RNA code of our threeenzymes In the other direction, “upstream”, it transcribes the neigh-boring “Regulator” sequence

b) Regulator Sequence

The Regulator sequence is the code of a fourth protein: lac repressor.

The lac repressor, which is not involved in the metabolic of lactose,

plays a key regulatory role in turning the gene “on” or “off”

c) Operator SequenceThe operator sequence is a DNA sequence that is recognized by lacrepressor If lac repressor is bound to the operator sequence, thendownstream gene expression is blocked The Figure 2 shows how this

“genetic switch” works

First, assume that the concentration of lactose in the environment is high

Lactose molecules bind reversibly to the repressor protein For high lactoseconcentrations, the lactose-bound form is favored under conditions of chem-ical equilibrium In the lactose-bound (or “induced”) form, the repressor

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Fig 2 The lac operon and gene regulation.

has a different structure in which it does not bind to the operator sequence.

RNA polymerase proteins binding to the promoter sequence are free to scribe in the down-stream (and up-stream) direction Along the downstreamdirection, it will produce an RNA copy of the genes of the three enzymesrequired for lactose breakdown Transcription along the up-stream directionproduces an RNA copy of the lac repressor gene Production of repressorproteins at a low level is necessary to maintain their concentration sinceproteins have a finite lifetime (after a certain period, a protein receives amolecular “tag” targeting it for future breakdown as part of the scheduledmaintenance program of the cell)

tran-Next, assume that the lactose concentration has dropped The chemicalequilibrium now favors the lactose-free conformation of the repressor Lac

repressor binds to the operator sequence and downstream gene transcription

is blocked Genetic switches of this type are used by E.Coli (and otherbacteria) to respond to changes in temperature, salinity, acidity, and theoxygen level Efficiency of these switches clearly is a matter of life anddeath for the bacterium so we should expect that the structure of proteinslike the lac repressor has been “sharpened” by natural selection for optimalperformance If you would put yourself the task of designing a lac repressorprotein some obvious minimum engineering requirements would be:

Specificity: the lac repressor must be able to recognize the operator quence Repressor proteins must be able to efficiently “read” the DNAcode

se-Reversibility: the lac Repressor must bind reversibly to lactose or else geneexpression could not be turned off Similarly, it must bind reversibly toDNA or else gene expression could not be turned on

Reactivity: The lac Repressor must locate the operator sequence withinminutes after the lactose concentration drops If it takes too long to turn agenetic switch then the bacterium could be dead before it had the change

to respond to the changing environment

In the next sections, we will see what thermodynamics, statistical ics, and elasticity theory have to say about these requirements First, wehave to learn more about the molecular structure of the two biopolymerfamilies [2]

mechan-1.2 Molecular structure

1.2.1 Chemical structure of DNAThe basic monomer unit – the polymer repeat unit – of double-strandedDNA is shown in Figure 3

The parts marked B and B∗are large, planar organic groups consisting of

one or two 5-atom aromatic rings They resemble benzene and, like benzene,

these groups do not dissolve very easily in water The symbols B and B∗

stand either for the smaller single- ring Cytosine and Thymine (the idines”), or the larger two- ring Guanine and Adenine (the “purines”) Wewill use the notation G, T, C, and A for short The four groups all have the

“pyrim-chemical character of a base (i.e., they are proton acceptors).

Not every combination of bases is permitted: in particular only B-B∗

pairs of purines and pyrimidines are possible The Watson–Crick pairing consists of combining A with T and G with C An A-T pair is

base-connected by two hydrogen bonds and a G-C pair by three hydrogen bonds,

so they have a higher binding energy Other purine-pyrimidine pairings

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Fig 3 Double-stranded DNA repeat unit.

(like G with T) are possible, but they do have a lower binding energy Thegenetic code of an organism, or “genome”, is simply a listing of the differentbase-pairings along the DNA sequence of that organism Note that if youknow the sequence of bases of one strand, you always can reconstruct theother “complementary” strand, assuming that Watson–Crick base pairing

is valid

The bases are connected to sugar groups (indicated by S in the figure)

Sugars have the general formula (CH2O)n and usually are water soluble

The particular sugar of DNA belong to the group of pentoses, 5-atom sugar rings, and is known as deoxyribose The deoxyriboses of two adjacent bases are connected together by tetrahedral phosphate groups (PO −

4) to form gether the sugar-phosphate “backbone” Adjacent sugar groups are sepa-rated by 6 ˚A The backbone strands have a directionality: they start with

to-a deoxyribose to-at the 3 end and end with a phosphate at the 5 end Thebackbone has two important physical characteristics for our purposes: it is

highly flexible and, in water at room temperature, it is highly charged The

negative charge of the backbone is due to the fact that the phosphate groups

in water at physiological acidity levels are fully dissociated Charged ular groups are usually soluble in water and the sugar-phosphate backbone

molec-is indeed highly soluble in water The flexibility molec-is due to the fact that thecovalent P-O bonds can freely rotate around so adjacent PO−4 tetrahedraand ribose rings along the backbone can rotate around their joining axis

We can describe the backbone as a charged, freely jointed chain.

RNA molecules are similar to single stranded DNA molecules with twodifferences First, the base Thymine is replaced by another base, Uracil,and second, the sugar group has an extra OH group and is called a ribose.

Intermezzo: Hydrogen bonding and the hydrophobic force

Hydrogen bonding provides the binding mechanism between complementarybases Hydrogen bonding plays in general a central role whenever macro-molecules are dissolved in water The hydrogen bond is an electrostaticbond with a positively charged proton from one molecular group associat-ing with a negatively charged atom of another molecular group, usually anoxygen (O−), Carbon (C−) or nitrogen (N−) atom The cohesion of wa-ter is due to hydrogen bonding between water molecules, with the proton

of one water molecule binding to the oxygen of another water molecule

The characteristic energy scale of the hydrogen bond is of the order of the

thermal energy kBT , so it is a relatively weak bond At room

tempera-ture, a thermally fluctuating network of hydrogen bonds connects the watermolecules

Molecules such as alcohol that are easy to dissolve in water are called

“hydrophilic” while molecules, such as hydrocarbons, that are not soluble

in water are called “hydrophobic” [5] Hydrophobic molecules cannot beincorporated in the thermally fluctuating network of the hydrogen bonds

They are surrounded by a shell of water molecules that have a reducedentropy, since they have fewer potential partners for the formation of ahydrogen bonding network As far as the water molecules are concerned,the surface of a large hydrophobic molecule resembles the air-water surface,

which has a surface energy γ of about 70 dynes/cm We thus can estimate

the solvation free energy – the free energy cost of inserting a molecule in

a solvent – as the surface area of the hydrophobic molecule times γ If

we wanted to dissolve a certain number of hydrophobic molecules we couldreduce the total exposed surface area in order to minimize the free energycost by collecting the hydrophobic molecules in dense clusters This effect

is known as the “hydrophobic interaction”, though it obviously is not a

pair-wise interaction between molecules Ultimately, the clustering leads tophase-separation, which you can observe when you try to mix oil with water

An important thermodynamic characteristic of the hydrophobic interaction

is that it is predominantly entropic in nature

1.2.2 Physical structure of DNAThe physical structure of double-stranded DNA is determined by the factthat it is neither hydrophobic nor hydrophilic It belongs to a special

intermediate group the “amphiphiles” that share properties from both

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double-helical structure of DNA To see how, imagine DNA stretched outlike a (straight) ladder (Fig 4)

Fig 4 Geometry of stretched DNA.

It turns out that the gaps between the rungs of the ladder, 2.7 ˚A, arewide enough to allow water molecules to slip in between the bases Underthe action of the hydrophobic force, the bases attract each other The fixed

6 ˚A spacing between the sugar groups prevents a contraction of the ladder,but there is another way to bring the bases in contact Imagine that we

gradually twist the ladder, thereby forming a double spiral This is possible

because of the flexibility of the backbone As we increase the twisting, thebases are brought into closer contact and the water molecules are squeezed

out For a twist angle T of about 32 degrees between adjacent bases, the

gap is completely closed This produces the classical double helix shown

below The repeat length is 360/T bases, or about 11 bases The repeat

distance along the helix, or pitch, is about 35 ˚A (using Fig 4, compute T

yourself)

The DNA double-helix is thus held together by the hydrophobic traction between bases, sometimes called the stacking interaction, and thehydrogen bonding between complementary bases The double-helix is notvery stable If you heat DNA, the two strands start to fall apart for temper-atures in the range of 70–80◦C In addition, a number of different variants

at-of the double helix can be realized by modest changes in the tal conditions Under conditions relevant for the life of cells, the dominantstructure is the “B form”, a right-handed helix with a 24 ˚A diameter In-creasing the salt concentration somewhat weakens the electrostatic repulsionbetween the two backbones A new structure, known as “A DNA”, appears,

Fig 5 B DNA.

with a smaller 18 ˚A diameter for the double helix and larger pitch of about

45 ˚A This structural flexibility of DNA is actually essential for its function:

in order for the genetic code to be “read” by RNA Polymerase and otherproteins, you must be able to “open-up” the double-helix Storing the ge-netic code in an overly rigid and stable storage device would be like having

a library with no doors

1.2.3 Chemical structure of proteinsThere are 20 different monomer units, or “residues”, that can be used to

construct protein biopolymers These are the naturally occurring acids Amino-acids have the form of a tetrahedron with a Carbon atom atthe center, denoted by Cα Recall that carbon has 4 electrons available tomake chemical bonds For the central Cαatom, these four electrons occupyfour electronic orbitals (the “sp4” orbitals) directed towards the vertices of atetrahedron (as in diamond) At the four vertices are placed, respectively, ahydrogen atom, an NH2“amino” group, an acidic COOH “carboxyl” group,

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the amino and hydroxyl groups are exchanged, we obtain the right-handed

or R form In proteins, only the L form is encountered The side groupdetermines the chemical character of an amino-acid It can be neutral orcharged, hydrophobic or hydrophilic, acidic or basic

Fig 6 Amino-Acid.

The simplest case is Glycine, with R a hydrogen atom There are twoabbreviations for Glycine: Gly and G Each amino acid has two of theseabbreviations (which biochemists know by heart) Lysine for instance is

a positively charged amino-acid with R equal to (CH2)4NH+

2 having theabbreviations Lys and K

Nature uses its own abbreviation for the 20 amino-acids when it storesthe information required to produce a protein Three adjacent DNA basescode for one amino-acid For instance, the triplet AAA is the code for theamino-acid “Phenylalanine” while TTT is the code for Lysine We call such

a triplet a “codon” You can construct 43= 64 different codons from such

a triplet, more that enough for the 20 natural amino-acids Finding thecomplete set of codons of all the amino-acids was one of the great landmarkachievements of molecular biology

To construct a protein, we must hook together these amino-acids by apolymerisation reaction This takes place between the amino group of oneamino acid and the hydroxyl group of another amino acid creating a covalent

bond – known as a peptide bond – between a Carbon and a nitrogen atom

under release of a water molecule The link between amino-acid complexes isnot rigid: the peptide bond allows considerable freedom of motion When

repeated over and over, this reaction produces a flexible string of aminoacids – a polypeptide – that starts with an amino group, the so-called “N-terminal” and that ends with a hydroxyl group, the “C-terminal”.

We saw that the base-pair sequence of the DNA of an organism is acode for the production of amino-acid strings with three adjacent base-pairs coding for one amino-acid The following DNA sequence for instancewill produce a simple polypeptide of four amino-acids, starting with an Nterminal and ending with a C terminal:

Fig 7 Codons and amino-acids.

(“Ala” stands for “Alanine”, a hydrophobic amino-acid.) Note that onlyone of the two DNA strands is actually used for the production of proteins,the “coding” strand

1.2.4 Physical structure of proteinsThe physical structure of protein is determined by two physical mecha-nisms On the one hand, proteins are again amphiphiles Among thestring of residues making up a protein, certain will have side chains thatare hydrophobic, like Ala, and certain that are hydrophilic, like Lys Whendissolved in water, the string will try to fold up into a ball, with the hy-drophobic residues hidden in the interior and the hydrophilic residues onthe exterior surface Such a ball is called a “globule”, with a radius of theorder of 2–3 nm

The second important effect is the ability of amino-acids to establishhydrogen bonds The oxygen atom of the C = O group at one of the corners

of the amino-acid tetrahedron of one residue can act as a proton receptorfor the C–H or N–H group of another residue Linus Pauling first proposed

that for a helical polypeptide string having the right pitch and diameter, known as the α-helix, every residue can establish a hydrogen bond with a

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Fig 8 A small protein with twoα-helices and a β sheet.

residue further up (or down) the helix Not every residue of a protein “likes”

to be part of an α-helix because certain side groups may interfere with each

other by steric hindering Hydrogen-bonding also can be used to link twostraight polypeptide strands that run either parallel or anti-parallel, which

is known as a β-sheet.

The actual structure of a protein is mainly determined by the combined

effects of α-helix/β-sheet formation and the requirement to keep

hydropho-bic residues inside the protein interior Drawings of protein structures show

the α-helical regions as spirals and the β sheets as arrows Figure 8 shows

an example of a very simple protein with two α helices and a β sheet Note that not all amino acids are part of α helices or β sheets.

What is remarkable about natural proteins compared with a randompolypeptide chain, is that over a certain range of temperatures, the mini-mum free energy state is a unique, folded structure with most of the atoms

of the protein occupying well-defined positions Only in this folded statecan proteins act as molecular machines This functional, folded state of

a protein is not very stable Formation of the folded structure involves asignificant loss of entropy Heating indeed unfolds, or “denatures” proteins

The “folding energy” – the difference between the properly folded state and

the denatured state – is only of order 10 kBT or so Moderately raising (or

lowering!) temperature, changing salt concentration or acidity level can duces unfolding However, it is precisely the fragility of folded proteins thatallows them to adopt multiple configurations, which permits their use as

switching devices, catalysts, and detection devices For a molecule to act

as a molecular machine, it must have “moving parts”

The following website has a nice tutorial on the chemical structure ofDNA and proteins

http://www.clunet.edu/BioDev/omm/exhibits.htm#displays

2 Thermodynamics and kinetics of repressor-DNA interaction

2.1 Thermodynamics and the lac repressor

The first branch of physics that we will bring to bear on the design of arepressor protein is thermodynamics/statistical mechanics We will apply

the principles of thermodynamics to understand how the specificity and versibilityrequirements are met for the interaction between the lac repressorand DNA

re-2.1.1 The law of mass actionPrepare an aqueous solution containing a certain low concentration of short,identical DNA strands and the repressor proteins The base-pair sequence

of the DNA strands may or may not contain the operator sequence We candescribe the reversible binding of the repressor to the DNA as an associativechemical reaction:

R + DNA ↔ R|DNA where R|DNA stands for a repressor-DNA complex The concentration of

DNA strands with no repressor is indicated by [DNA], that of free

repres-sor by [R], and that of the complexes by [R|DNA] Concentrations can be

measured by filtration methods and the results are expressed in “Molar”, ormoles per liter (symbol M) Salt water has, for instance, a salt concentra-tion of about 0.1 M while one molecule per micron3(volume of a bacterium)equals 10−9 M

Thermodynamic processes inside cells normally take place under tions of (nearly) fixed temperature and pressure Under these conditions,the Gibbs Free Energy G must be minimized according to the Second Law

condi-of Thermodynamics The Gibbs Free energy can be expressed as:

G = NDNAµDNA+ N R µ R + N R|DNA µ R|DNA (2.1)

Here, N and µ are the number of molecules and the chemical potential of

each of the three species (actually, we also should add a term for the water

molecules) At low concentrations, the chemical potential µ([C]) of “solute”

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The first term, with νCthe volume of the molecule, is very similar to the freeenergy per particle of an ideal gas and it is indeed due to the translationaldegrees of freedom of the solute particles The second term, the “standardchemical potential”, can be viewed as the intrinsic free energy per soluteparticle meaning that it depends on the type of solute molecule, and ontemperature and pressure, but not on concentration

Assume that there is a very small variation δN in the number of R|DNA complexes According to the reaction scheme R + DNA ↔ R|DNA, there must be corresponding variation of −δN in the number of uncomplexed DNA and repressor molecules The change in G equals:

δG = [µ([R|DNA]) − µ([DNA]) − µ([R])] δN. (2.3)

The Second Law of Thermodynamics demands that δG = 0 so µ([R|DNA])

= µ([DNA) + µ([R)] Using equation (2.2) and this condition gives:

can bind to DNA

Equation (2.4) is a special case of a fundamental principle of chemical

thermodynamics: the Law of Mass Action The Law of Mass Action is such

an important principle that the right hand side of equation (2.4) has it’s

own name and symbol: the equilibrium constant Keq

Equilibrium constants of associative reactions have dimensions of tration, so they are expressed in Molar Using the Law of Mass Action, the

equilibrium constant can be obtained by measuring the concentrations, andhence the standard free energy change The beauty is that we obtain thisway an important microscopic quantity, the standard free energy change,

by measuring purely macroscopic quantities

When such an experiment is performed in a test-tube (“in vitro”) on a

DNA/repressor solution [6], one finds that the result is very sensitive to theabsence or presence of the operator sequence on the DNA:

Keq≈

This large difference between the specific and non-specific equilibrium stants is the thermodynamic signature of the ability of repressor proteins to read DNA sequences

con-We call the interaction between lac repressor and operator DNA the

“specific” protein-DNA interaction and that with non-operator DNA the

“non-specific” interaction You might expect the equilibrium constant forthe non-specific interaction to be independent of the DNA sequence but

it actually can vary over two orders of magnitude when the non-operatorsequence is varied Later, this will turn out to be a quite important effect

From equations (2.6) and (2.7), one finds that the standard free energy

change for the operator case ∆G0 (specific) is of the order of

20−25 kBT while for the non-operator case ∆G0 (non-specific) is of the

order of 5−10 kBT

What happens if we apply the Law of Mass Action to conditions relevant

to the crowded interior of E.Coli (rather than test-tubes)? The genome ofE.Coli contains about 107base-pairs (or “bp”) restricted to a volume of the

order of one µ3 Let’s approximate the non-operator part of the bacterial

genome as a fairly concentrated solution of short (10 bp) DNA sequenceshaving a concentration of the order of 106/µ3(10 milliMolar) First supposethat the lac repressors all are bound to lactose molecules, so they will notrecognize the operator sequence Let F be the fraction of unbound lacrepressors This “free fraction” can be related to the equilibrium constantthrough the Law of Mass Action:

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specific equilibrium constant Keq is measured in the absence of lactose so

we are assuming here that lactose binding does not affect the non-specificinteraction) That is an interesting result! Induced lac repressors in E.Colistill “live” most of the time on DNA even though they do not recognize theoperator sequence

Intermezzo: Energy scales in molecular biochemistry

This 25 kBT value for ∆G0 (specific) is a typical energy scale for thecomplexation of biological macromolecules On the one hand, this en-

ergy scale must be sufficiently high compared with the thermal energy scale kBT so thermal fluctuations do not break up the complex On the

other hand, the energy scale must be sufficiently low so the binding is versible and can be easily disrupted when required for the signaling process

re-In molecular biology, the universal “energy currency” for driving dynamically unfavorable processes is the hydrolysis of an ATP molecule:

thermo-ATP +H2O → ADP + Pi + H, which delivers about 10 kBT in free energy.

A 25 kBT value for the binding energy is thus quite reasonable Protein

complexes are in general maintained by multiple “weak bonds”, such asthe van der Waals attraction, hydrogen bonds, and the “polar” interaction

(i.e screened electrostatic interaction), all of the order of kBT Spatial

patterns of these weak links provide a basis for highly specific key” type recognition between proteins This must be contrasted with the

“lock-and-covalent “strong bonds” (of the order of a hundred kBT ) that maintain the

structural integrity of the macromolecules

2.1.2 Statistical mechanics and operator occupancyNow assume that the lactose concentration has dropped so the lac repres-sor proteins can bind to the operator sequence Efficient design requires

a high probability for the operator site to be occupied (to avoid unwanted

gene transcription) We will compute the operator occupancy probability P using elementary statistical mechanics Let there be M copies of the lac re- pressor distributed over N possible sites of the bacterial genome (with N,

of the order of 107, large compared to M) We will neglect the small tion of free repressors There are then A(N, M) = N.(N − 1) (N − M) ways to distribute the M proteins over the N non-operator sites and there are C(N, M) = M[N.(N − 1) (N − (M − 1))] ways of choosing one of the M proteins to occupy the operator site and distribute the remain- ing M − 1 proteins over the non-operator sites, treating the proteins as

classical, distinguishable objects Let the Boltzmann factor of a protein

occupying an operator site, respectively, a non-operator site, be B s,ns ≡ exp(+∆G0(specific, non − specific)/kBT ) The occupation probability is

then

P = C(N, M)Bs(Bns)M−1

C(N, M)Bs(Bns)M−1 + A(N, M)(Bns)M · (2.9)This simplifies to

1 +N M



where ∆∆G0 = ∆G0 (specific)−∆G0 (non-specific) is the difference tween the specific and non-specific binding energies [7]

be-When we put in the “numbers” for the binding energy obtained earlier

something interesting shows up: the very large number N and the very small number exp(−∆∆G0/kBT ) nearly cancel each other (N exp(−∆∆G0/kBT )

is about 10) Suppose we wanted to make sure that the operator is at least99% of the time occupied According to equation (2.10), that requires the

number of copies M of the lac repressor to exceed 103 The actual number

of lac repressors of an E.coli bacterium is maintained at a comparable value(about 102) There is thus a “design connection” between the values of thespecific and non-specific binding energies on the one hand and the number ofrepressor copies maintained by the cell on the other hand Simple statisticalmechanics arguments provide us with insight how the “working parameters”

are set for bacterial gene expression The most important lesson is that the

value of quantities such as ∆G0 (specific), ∆G0 (non-specific), N and M

must be understood in the light of the functioning of the bacterium as an

integrated system.What is puzzling at this stage is why we need the non-specific interaction

in the first place According to equation (2.10), if we turned off the specific interaction, we would only need about 10 repressor copies We willreturn to that question in the discussion of the kinetics

non-2.1.3 Entropy, enthalpy, and direct read-outThe Gibbs Free Energy is defined as

It is the sum of an “energetic” term: the enthalpy H = E + P V (E is

the internal energy) and an “entropic” term The change in Gibbs Free

Energy ∆G0 that takes place when a repressor molecules binds to DNAcan be obtained from the equilibrium constant Can we obtain the separate

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with ∆Q the heat released (which is why we call the enthalpy also the

“heat function”) We call chemical reactions exothermic if ∆Q > 0 and endothermic if ∆Q < 0 Endothermic reactions are interesting because the

driving mechanism is entropy increase rather than reduction of the potentialenergy of interaction between molecules The heat released by a reactioncan be measured by calorimetry so the change in enthalpy can be found

Since the total change of the Gibbs Free Energy is known, we can alsodeduce the change in entropy

When the enthalpic and entropic contributions ∆H and −T ∆S are

de-termined in this manner for the interaction between the lac repressor andDNA, one finds the following results [8]

Specific interaction

The dominant contribution to ∆G0 is entropic As a function of

tempera-ture, −T ∆S decreases significantly with T ∆H is negative so the reaction

is endothermic

Non-specific interaction

The dominant contribution to ∆G0 is again entropic, but −T ∆S now does

not depend significantly on temperature The enthalpic contribution isagain negative

Both are surprising results To see why, we turn to the results of structuredeterminations of protein-DNA complexes It is possible to grow crystals ofrepressor proteins complexed with short bit of DNA, known as “co-crystals”

X-ray diffraction experiments on these crystals allow us to determine atomicpositions with a resolution of 2A, and sometimes even better than that [9]

Below we show the result of such an experiment for case of “cro” a verysimple bacterial repressor (unlike the lac repressor)

The first panel shows the pattern of chemical bonds There is a C2

rotation symmetry This symmetry is a characteristic of many prokaryoterepressor proteins The DNA operator sequence has a corresponding (ap-proximate) rotation symmetry Simple repressor proteins like cro address

the DNA with “reading heads” A reading head is an α-helix that can be

inserted into the major or minor groove of the DNA double helix (usuallythe major groove) The second panel is a cartoon of the cro repressor/DNA

complex showing the α-helices of the protein There are two reading heads

Fig 9 Cro-repressor/DNA Complex First panel: chemical bonds Second panel:

cartoon showing reading heads

visible, one near the top and one near the bottom The ends of certain sidechains of the reading head can establish specific links with certain DNAbases An example is the interaction between the amino-acid Arginine andthe base Guanine shown in Figure 10 below

The Arg side chain terminates with two N-H pairs The two hydrogenatoms are positively charged and they “fit” exactly with negatively chargednitrogen and oxygen atoms of the Guanine base The nitrogen and oxygenatoms act as proton acceptors so hydrogen bonds can be established, indi-cated in the figure by the two ovals Base-pairs are surrounded by a uniquecombination of proton donors and proton acceptors that can be read byspecific amino-acids For instance, the amino-acid Glutamine “recognizes”

an A-T pair in the major groove of DNA, just as Arginine recognizes a G-Cpair in the major groove, while Asparagine recognizes a G-C pair in theminor groove

We call this the “Direct Read-Out” mechanism [10] and it is based onhydrogen bonding between amino-acids and nucleic acids

Intermezzo: The second code

Molecular Biologists have established long lists detailing contacts betweenthe amino-acids of DNA-binding proteins and DNA base-pairs [11] Theyoriginally hoped they could determine a “second code” By this they mean

a one-to-one relation between amino-acids and base-pairs so they could dict to which base-pair sequence a given repressor protein would bind Thatwould enable design of highly specific drugs turning on or off particular

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Fig 10 Direct read-out.

genes Unfortunately, there appears to be no universal second code associating proteins come in different design forms The same amino-acidsinteract differently in different types of proteins

DNA-There is an obvious discrepancy between the Direct Read-Out model andthe results obtained from thermodynamics If hydrogen bonding betweenthe reading heads and DNA really was the dominant binding mechanism,then DNA/repressor binding should have been enthalpic in nature and for-mation of the complex would be associated with a loss of entropy Thepuzzle is that there can be little doubt that Direct-Read Out is an impor-tant mechanism for the reading of DNA sequence by proteins

2.1.4 The lac repressor complex: A molecular machineThe resolution of this paradox comes from X-ray structural studies of lacrepressor/DNA co-crystals [12] shown below

The actual structure responsible for the repression of gene tion is a complex consisting of two lac repressor protein dimers, so fourcopies in all They bind pair-wise to two separate operator sequences; notethe four reading heads The four reading heads are pair-wise attached to

transcrip-the body of transcrip-the complex by a linker unit that undergoes an order-disorder

Fig 11 The lac repressor complex.

transition upon lactose binding In the presence of lactose, the complexadopts a structure in which the linker unit is disordered and the readingheads can not be inserted into the DNA major groove Release of the lactoseproduces ordering of the linkers and allows insertion of the reading heads

into DNA In addition, the transition brings two hydrophobic surfaces,

be-longing to the two dimers, into close contact It seems reasonable to assumethat if lactose-free repressor monomers or dimers move along non-operatorDNA, locate the operator sequence, and form the full four-protein repressorcomplex, then the hydrophobic attraction plays a central role as well, so wecan understand at least qualitatively why the specific binding of the lac re-pressor has an entropic character The intervening DNA sequence betweenthe two operator sequences loops around as shown in Figure 12 Interest-

ingly, another protein, known as CAP, binds to the DNA sequence inside

the loop This stabilizes the loop but once the loop opens, it also stimulatesgene expression!

The reading heads thus are only a small part of the lac repressor

com-plex We could view the complex as a molecular detector and amplifier.

The binding of lactose to the repressor complex triggers a large structuraltransition that breaks up the complex and opens the loop Release of thelactose closes the loop and restores the complex Note that there is an anal-ogy between the operation of the lac repressor complex and the molecular

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Fig 12 Order-disorder Transition of the lac repressor complex.

The thermodynamics of the non-specific interaction also is puzzling Thebinding free energy had a different temperature dependence, indicating thatthe hydrophobic interaction is not the interaction that dominates the ther-modynamics, which is indeed the case The structural studies tell us thatthe linker units connecting the reading heads of the lac repressor to the

body of the protein are positively charged and interact with the adjacent

minor groove: the non-specific interaction is electrostatic in origin We mustunderstand how electrostatic attraction can have an entropic character Wewill postpone addressing this question to Section 4

At this point, you should go the following Web site where you will find

an elegant tutorial on the structural changes of the lac repressor tetramerand its interaction with DNA

http://www.worthpublishers.com/lehninger3D/index title.html

2.2 Kinetics of repressor-DNA interaction

We now turn to the third engineering requirement: reactivity How quickly

does a lac repressor respond to environmental changes, such as a

reduc-tion in lactose concentrareduc-tion? We start again with a discussion of in vitro

experiments

2.2.1 Reaction kineticsThe rate of change with time of the concentration of a repressor-DNA com-plex is the sum of two terms A positive contribution due to complex for-mation between a previously unbound DNA molecule and a previously freerepressor, and a negative contribution due to complex break-up At suffi-ciently low concentrations, the first term must be proportional to the prob-ability of finding a free DNA molecules and a free repressor molecule at thesame site, and the second term must be proportional to the concentration

of the complex:

d

dt [R|DNA] = ka[R][DNA] − kd[R|DNA]. (2.13)

The proportionality constants ka and kd are called, respectively, the rate ” and the “off-rate” These constants are supposed not to depend on

“on-concentration though they can be quite strongly temperature dependent

The off-rate really does have dimensions of a rate but the (so-called) rate has dimensions of Volume/Time (chemists and biologists have a free-and-easy attitude to units) The on-rate and the off-rate have a surprisingconnection Under conditions of thermodynamic equilibrium, the concen-trations of the reactants obviously must be constant, so the left hand side

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rela-equation (2.15) In vitro experiments on repressor-DNA solutions ing the operator sequence) report that for the lac repressor ka is of order

(contain-1010M−1 s−1 under standard conditions

Using this information, let’s apply equation (2.13) to a colony of E.Coli

bacteria Suppose that at times t < 0 there are no complexes because the environmental concentration of lactose is high At time t = 0, the

lactose concentration drops to zero How long will it take the activated lacrepressors to locate the operator sequence and switch-off gene expression?

There are only a few operator sequences per E.Coli Assuming a volume

of 1µ3, the (initial) concentration of unoccupied operator sequences is of

order 1/µ3 or about 10−9 M According to equation (2.13), for early times

t, the concentration of occupied operator sequences in the colony will grow

linearly in time as:

d

keeping in mind that at t = 0, [R|DNA] = 0 It follows that we can identify

τ = 1/(ka[DNA]) as the characteristic time scale for a free repressor to

locate the operator sequence, the switching time in other words For the measured value of ka, this switching time is of order 0.1 s This is a sensibleresult from the viewpoint of design: the actual switching time should beless than a minute or so for genetic switching to be a relevant response to achanging environment Our estimate of the switching time must be viewed

as a lower bound, because the cell environment is quite crowded The actual on-rate inside a cell must be significantly less than this in vitro value This means that the in vitro on rate must be of order 1010 M−1 s−1 (or higher)

for reasonable in vivo repressor reactivity.