What a concentration leaves out

An average can be correct and still leave the experiment undecided. The students' extensions ask which molecular details a model must retain.

CCBS · EST 5053 · Lecture 2 audience extension
Editorial synthesis of Week 1 student submissions · prepared with Codex · 8 September 2026

A concentration replaces the identities and positions of molecules with one number. A rate law replaces many events with their average production of change. Lecture 2 uses these compressions to make the cell calculable. Its scales tell us where they might work.

The Week 1 submissions put that permission under pressure. A molecule can fit without being accessible. A concentration can be nonzero while a particular cell is empty. A protein can reach its target without recognizing it. Two enzymes can give the same mean rate while executing very different numbers of microscopic events.

These are not four reasons to abandon simple models. They are four changes in what we want a model to predict. We will follow that change of question from molecular volume to populations, from populations to spatial search, and from mean reaction rates to the events and balances hidden underneath them.

Student names link to the submitted arguments at their points of use. New connections and calculations are editorial synthesis, not quotations or an invented classroom dialogue. This audience essay extends the core, with its exposition and lineage. The lecturer's receptor-capture example remains separate.

1Which volume?

The volume of molecular material is not the volume of the region a molecule occupies. That distinction matters before we decide whether the cell is spacious, crowded or well mixed.

Yibing Li (李艺冰) tests the apparent packing problem posed by a bacterial chromosome. Take one 4.6-million-base-pair genome, a rise of 0.34 nm per base pair, and a bare DNA radius of 1 nm. The contour is about 1.56 mm long, yet its cylindrical material volume is only

Since $1\,\mathrm{fL}=1\,\mu\mathrm{m}^3$, that is about 0.49% of a chosen 1 fL cell. The millimeter-long contour is not a contradiction. It is the length of an exceptionally thin object.

Yibing's estimate answers a material-volume question. It does not measure the nucleoid's volume. A folded polymer occupies a region that includes solvent and gaps. DNA-binding proteins, hydration and multiple genome copies add further distinctions. A low bare-material fraction therefore does not imply that another molecule can move through every part of the chromosome equally easily.

Álvaro Bátrez extends the bookkeeping from DNA to its RNA copies. For the coding region of an mRNA, three nucleotides encode one amino-acid residue. Using approximate masses of 330 Da per nucleotide and 110 Da per residue gives a message-to-product mass ratio of $3(330)/110\simeq9$. Untranslated regions add RNA mass.

A heavy message can still be useful if it is reused. But the number of proteins produced from one message depends on its lifetime and translation rate. Mass alone cannot establish that yield, its energetic cost or the polymer's spatial envelope. Álvaro's question therefore identifies the next quantities to measure rather than a universal penalty for RNA.

The same discipline will matter for concentration. A compact number is useful only after we know which physical quantity it represents.

Distinguish bare material volume, spatial occupancy and copy-production cost before inferring what fits or what is accessible.

2Which population?

A concentration-to-count conversion produces an expected number, not a fraction of a molecule in one cell. Zixuan Wang (王子轩) makes that distinction quantitative by asking what 1 nM means inside 1 fL.

Let $\lambda$ be the mean count. Avogadro's constant and the unit conversions give

At 1 nM in 1 fL, the mean is 0.602 molecules. To predict how often a cell is empty, however, we need more than that mean. We need a distribution.

A simple population model supplies one. Molecules enter at a constant rate, leave independently at a rate proportional to their number, and have reached a stationary distribution. Under those assumptions the count $N$ is Poisson:

The empty-cell probability is then about 0.548. More than half the instantaneous samples are empty even though the ensemble concentration is 1 nM. The standard deviation is $\sqrt{\lambda}$, so the coefficient of variation, defined as standard deviation divided by mean, is $\mathrm{CV}=1/\sqrt{\lambda}$.

Zixuan's result separates two design requirements that “enough molecules” might conceal. Requiring at least one molecule with 99% probability needs a mean of at least $-\log(0.01)=4.605$. Requiring an instantaneous CV no greater than 10% needs a mean of at least 100. Reliable presence and a precise population are different objectives.

Increasing volume at fixed concentration raises the mean count. Averaging over time is a different intervention. Repeated samples help only according to how much new information they contain, so their correlation time matters. The first question for a low-copy assay is consequently whether it measures an ensemble mean, an instantaneous count or a time-integrated signal.

Convert concentration into mean count, then state a population model before predicting absence or relative noise.

3Faster than what?

Large copy number does not remove spatial structure. To average positions away, we must compare the time needed to explore a relevant distance with the time over which reaction changes the population.

Changkai Li (李常凯) compares these two clocks. Consider two equally mobile partners. Their relative diffusion coefficient is the sum of their individual coefficients, $D_{\mathrm{rel}}=D_1+D_2$. For a chosen relative-mixing distance $L$, use the scale $\tau_{\mathrm{mix}}\sim L^2/D_{\mathrm{rel}}$.

The reaction clock needs a capture model. For an isolated perfectly absorbing spherical encounter surface of radius $a$, the volume-rate coefficient is $4\pi D_{\mathrm{rel}}a$. If the partner's number density is $\rho$, a tagged molecule's pseudo-first-order rate is $4\pi D_{\mathrm{rel}}a\rho$. The ratio of the two clocks is therefore

The symbol $\mathrm{Da}$ denotes a diffusion–reaction timescale ratio. The diffusion coefficient cancels because this particular model uses the same mobility for both processes. Slowing all motion equally does not automatically make mixing fast relative to capture.

Changkai uses the GFP diffusion scale measured by Elowitz and colleagues. Their value of about $7.7\,\mu\mathrm{m}^2\mathrm{s}^{-1}$ is a measurement for that protein and experimental setting. Assigning it to both partners gives an illustrative $D_{\mathrm{rel}}=15.4\,\mu\mathrm{m}^2\mathrm{s}^{-1}$, not a measured mobility for every binding pair.¹

With $L=1\,\mu\mathrm{m}$ and encounter radius $a=4\,\mathrm{nm}$, the mixing scale is about 0.065 s. The perfect-capture association constant is about $4.66\times10^8\,\mathrm{M}^{-1}\mathrm{s}^{-1}$. The two clocks become comparable near 33 nM.

A ratio near one removes this simple argument for fast global mixing. It does not prove that every mass-action prediction fails. A spatially uniform population may still admit a useful mean-rate description, while a localized source or sink may require positions. The relevant comparison depends on the observable and its spatial scale.

Lipei Fang (方利培) introduces a second change: crowding can affect both mobility and what happens at contact. Once capture is imperfect, a reduced association rate need not track the large-scale diffusion coefficient. Excluded volume and interactions can also favor association in some regimes while transport slows. A single viscosity correction cannot stand in for every effect. The crowding analysis of Zhou, Rivas and Minton makes these competing contributions explicit.²

The next useful comparison is therefore measured mobility alongside measured association, at the scale the model will use. That also forces us to say what the capture surface actually represents.

Compare mixing and reaction on a stated length scale, and treat the resulting separation as an assumption to test rather than a universal cellular property.

4What is the target?

A target radius is a property of a capture geometry, not a number to borrow from any nearby molecular length. Chengwei Jin (金城炜) asks which length belongs in a DNA-binding estimate: the protein radius, the operator's contour length, or the effective size of an absorbing patch.

The distinction is visible in Berg and Purcell's small-patch calculation. A circular absorbing patch of radius $s$ on an otherwise reflecting surface has volume-rate coefficient approximately $4Ds$ in the local planar limit. Here $s$ is the patch radius. It is not the radius of the whole cell or the hydrodynamic radius of the searcher.³

Take $D=50\,\mu\mathrm{m}^2\mathrm{s}^{-1}$ and $s=1\,\mathrm{nm}$ as an illustrative geometry. Converting the volume rate to molar units gives about $1.20\times10^8\,\mathrm{M}^{-1}\mathrm{s}^{-1}$. Riggs, Bourgeois and Cohn reported a lac repressor–operator association constant of $7\times10^9\,\mathrm{M}^{-1}\mathrm{s}^{-1}$ in their 1970 assay. That is roughly 58 times this chosen patch prediction.⁴

We can ask what changing only the radius would require. Because the patch coefficient is linear in $s$, matching the reported rate would require about 58 nm. This is not a measurement of the operator. It shows the cost of insisting that the simple patch model explain the observation by geometry alone.

Chengwei's question stops us from disguising that cost with a convenient length substitution. An operator's base-pair contour length does not automatically define a circular absorbing surface. The mismatch invites additional mechanisms, including electrostatic effects and search through nonspecific DNA association. A rate discrepancy alone does not uniquely select one of them.

Identify each geometric length by its role in the capture problem before using a rate mismatch to infer a new mechanism.

5When should a searcher let go?

DNA association creates a search tradeoff. Remaining attached permits local exploration. Detaching costs time but can move the searcher to a new region. Siheng Wang (王思恒) asks how long local exploration should continue before that relocation becomes worthwhile.

Define a renewal-search model. A round consists of a three-dimensional relocation with mean duration $\tau_3$, followed by one-dimensional sliding that explores a fresh length $\ell$ of DNA. Let $D_1$ be the sliding diffusion coefficient in length-squared per time. Approximate the sliding duration by $\tau_1=\ell^2/(2D_1)$.

If the genome has search length $G\gg\ell$, each round samples a fresh effective region, and encountering the target guarantees recognition, the mean search estimate is

Longer sliding reduces the number of relocations but increases the total sliding cost. Differentiating gives $\ell_*=\sqrt{2D_1\tau_3}$. At this optimum, the per-round sliding and relocation times are equal. With $x=\ell/\ell_*$, the normalized prediction becomes $T/T_{\min}=(x+x^{-1})/2$.

Experiment gives the model a useful objection. Hammar and colleagues inferred a sliding length of $45\pm10$ base pairs for lac repressor in living bacteria using a model that includes target recognition. They also found that the repressor often passes the natural operator without binding it. Reaching a sequence and recognizing it are separate events.⁵

This distinction changes the optimization problem. The same paper shows that imperfect recognition removes the universal equal-time optimum. A longer DNA-bound fraction can be favorable because additional passes create opportunities for recognition. Siheng's calculation has not become useless. Its clean optimum tells us exactly which assumption the recognition experiment tests.

The next measurement should distinguish relocation, sliding and recognition, rather than compress them into one association constant. A search-time curve alone may fit several combinations of those processes.

Derive a search optimum together with its renewal and recognition assumptions, then choose an experiment that tests those assumptions.

6Stronger for which task?

Finding and occupying a target is not the same task as distinguishing two input concentrations. Once a target is almost always occupied, making binding stronger can leave less room for its occupancy to change.

Kehan Lang (郎可菡) turns this observation into an optimization problem. For a single equilibrium binding site, define the mean occupancy at concentration $c$ by $p(c)=c/(c+K_d)$. Here $K_d$ is the dissociation constant, and smaller $K_d$ means stronger affinity.

The chosen output is the difference in mean occupancy between two fixed inputs $c_1<c_2$:

This contrast vanishes at both extremes. Very weak binding leaves both inputs mostly unoccupied. Very strong binding leaves both mostly occupied. Setting the derivative with respect to $K_d$ to zero gives the intermediate optimum

For 10 and 20 nM, the optimum is about 14.1 nM and the contrast is 0.172. At 1 nM, stronger binding gives a smaller contrast of 0.0433. Kehan's calculation specifies a task for which stronger affinity is no longer synonymous with a better response.

A common phenomenological generalization is the Hill response, $p_h(c)=c^h/(c^h+K^h)$. Its positive exponent $h$ controls steepness, and $K$ is the half-occupancy concentration. Substituting $c_1^h$, $c_2^h$ and $K^h$ into the same contrast calculation gives $K^*=\sqrt{c_1c_2}$ again. This chosen response curve does not, by itself, specify a molecular binding mechanism.

The objective's name matters. Maximum mean contrast is not automatically maximum information, statistical discriminability or response speed. Those questions require fluctuations and kinetics. Turning binding-site occupancy into gene expression requires further assumptions about the regulatory mechanism, as the thermodynamic models of Bintu and colleagues explicitly explain.⁶

This result concerns two inputs at one site. If the objective becomes delivering molecules to many sites at different distances, strong capture can create another limitation.

State the response objective before optimizing affinity, and keep mean contrast distinct from noisy discrimination or speed.

7Binding can intercept delivery

Rapid local capture can reduce access farther from a supply boundary. Xinyu Wu (吴新宇) explores this binding-site barrier in antibody delivery: molecules captured near vessels are no longer freely available to diffuse deeper into tissue.

We can isolate the mechanism without fitting a tumor. Let $C(x)$ be free antibody concentration at distance $x$ from a planar source. Assume a constant diffusion coefficient $D$, a constant effective first-order removal coefficient $k$, and a maintained boundary concentration $C_0$. At steady state in a semi-infinite region,

Substituting an exponential gives the penetration length $\ell=\sqrt{D/k}$ and profile $C(x)=C_0e^{-x/\ell}$. The coefficient $k$ lumps removal processes. Its first-order form does not make the underlying uptake mechanism elementary.

This length gives the earlier timescale comparison a spatial consequence. Across a tissue distance $L$,

When this ratio is large, the model predicts strong depletion before the molecule explores that distance. Changkai's clock comparison has become a concentration profile because we have now specified a source, a sink and a boundary condition.

The published experiment tests a related intervention. Bordeau, Yang and Balthasar used a transient competitive inhibitor with trastuzumab in a mouse xenograft model. At 24 h, Table 2 reports mean distances of antibody-positive image pixels from vessels of $41.30\pm6.70\,\mu\mathrm{m}$ without the inhibitor and $58.24\pm5.40\,\mu\mathrm{m}$ with it. These are mean ± SD, with three animals and nine tissue slices per group.⁷

The image metric is not the exponential decay length of free concentration. Identifying the two would require an observation model connecting free antibody, bound antibody, staining thresholds and vessel geometry. The reported distance increase therefore cannot by itself establish a numerical reduction in $k$. Xinyu's mechanism motivates the model, while the experiment tells us which observable it must eventually predict.

Connect a diffusion–reaction ratio to a spatial prediction, while keeping a model's decay length separate from the metric an imaging experiment actually reports.

8One curve, many mechanisms

A rate curve can hide microscopic distinctions even when space has been averaged successfully. Yanqing Shi (施延庆, LeoSin) demonstrates this through stochastic simulation cost. Two descriptions can make product at the same mean rate while requiring radically different numbers of elementary events.

Resolve the enzyme into reversible binding and product formation. Let $E$ be free enzyme, $S$ buffered free substrate, and $C_{ES}$ their complex. The modeled elementary steps are

At a fixed substrate concentration, the stationary mean product rate per total enzyme concentration $q_E$ is

Fixing $K_M$ and $k_{\mathrm{cat}}$ fixes this curve, but not the two binding constants individually. We may vary $k_{\mathrm{on}}$ and choose $k_{\mathrm{off}}=K_Mk_{\mathrm{on}}-k_{\mathrm{cat}}$, provided the resulting dissociation rate is nonnegative. Briggs and Haldane already identified this limitation of the rate curve in their original two-page note.⁸

To count the hidden work, define $r=k_{\mathrm{off}}/k_{\mathrm{cat}}$. Each bound episode ends in product with probability $p=1/(1+r)$. It therefore takes $1+r$ binding episodes on average to make one product. Each episode contains a binding event and an exit event, either dissociation or catalysis:

The hidden difference can affect a physical observable as well as computation. For one enzyme at $S=K_M$, define the turnover interval $T$ from one product event to the next. Both mechanisms have mean $\operatorname{E}[T]=2/k_{\mathrm{cat}}$. Their normalized waiting-time variance, or randomness parameter, is

These are chosen parameter sets, not two measured enzymes or a matched “median enzyme.” Their stationary mean curves coincide at every buffered substrate concentration. Their startup transients and turnover distributions need not. A mean-rate measurement cannot distinguish them, but sufficiently resolved waiting-time data can.

Yanqing's simulation question thus becomes an experimental-design question. We should retain microscopic events when the observable needs them. If only the steady mean is required, a justified reduced description may be cheaper. The mean curve alone cannot establish which hidden rates are fast or which binding relation is near equilibrium.

Construct mechanisms with the same mean rate, then choose an event-count, transient or fluctuation measurement that can distinguish them.

9Balanced does not mean inactive

A final kind of hidden information is the circulation of shared molecular resources. Even a correct rate law for each isolated reaction is insufficient if those reactions depend on the same cofactor pool.

Runzi Yan (闫润梓) asks why oxygen availability can affect the TCA cycle even though its individual cycle reactions do not directly consume molecular oxygen. The useful connection is cofactor recycling: NAD-dependent oxidation produces NADH, while respiratory electron transfer can regenerate oxidized cofactor using oxygen as the final electron acceptor.⁹

Write the smallest balance that exposes the dependency. In one modeled compartment, let $H$ denote NADH concentration and $O$ denote NAD⁺ concentration. The letter $O$ here names oxidized cofactor, not molecular oxygen. On a short interval with negligible synthesis, loss and compartment exchange, assume their total is fixed:

Let $J_{\mathrm{red}}$ be the flux that reduces NAD⁺ to NADH, and $J_{\mathrm{ox}}$ the flux that reoxidizes NADH. Both have concentration-per-time units. Cofactor bookkeeping gives

A stationary pool requires $J_{\mathrm{red}}=J_{\mathrm{ox}}$. It does not require either flux to vanish. As in Anqi Tu's ATP ledger, an unchanged amount can support continuing turnover. Equal total production and consumption also do not establish thermodynamic equilibrium.

Now specify an intervention. If reduced oxygen availability lowers the relevant oxidation flux, and no alternative route compensates immediately, NADH initially accumulates and NAD⁺ falls. NAD⁺-dependent reactions then face a changed substrate environment. Oxygen can therefore influence a reaction through the network without appearing as that reaction's direct substrate.

Alternative regeneration routes are experimentally consequential. Sullivan and colleagues showed that an alternative electron acceptor increased the NAD⁺/NADH ratio and supported proliferation in respiration-deficient cultured cells. That is a concrete reason to keep the compensation clause in the model.⁹

The balance does not predict a universal oxygen threshold or simultaneous cessation of every TCA reaction. Those claims require enzyme kinetics, compartment-specific pools and alternative pathways. Runzi's question instead provides a direct entry into Lecture 3: write the coupled balances before treating one reaction's inputs as independently fixed.

Use a conserved cofactor pool to identify indirect dependencies, while distinguishing stationary amounts from zero flux or equilibrium.

10Choose what may be forgotten

Model resolution is a choice about observables. The audience's extensions have not selected one permanently correct level of detail. They have shown how a new measurement can make a previously useful average insufficient.

There are three distinct decisions. We may average over molecule number, over position, or over intermediate reaction events. A fourth obligation remains at every level: preserve the balances that couple the variables we keep. None of these permissions follows simply from the cell being small.

The cleanest next experiment is one for which two plausible descriptions agree on the old measurement and disagree on the new one. The enzyme pair supplies an explicit example. Measure only the stationary mean rate and the mechanisms remain indistinguishable. Measure a turnover distribution at a specified substrate concentration and they make different predictions.

That criterion also keeps extension work focused. We do not add DNA geometry, spatial fields or hidden states merely because the cell contains them. We add them when a question depends on them. Lecture 3 develops the coupled balances, Lecture 5 examines discrete events, and Lecture 6 asks when fast internal motion can be eliminated. The students' questions have supplied tests that those later methods will have to pass.

Choose the smallest model that distinguishes the outcomes your experiment is designed to distinguish.

Reproduce the numbers

The six figures are model calculations, not digitized biological data. This standard-library Python code gives their defining functions and the principal numerical checks. The experimental distances in §7 are transcribed from the cited table, not used to fit the plotted profiles.

from math import pi, exp, sqrt, log
NA = 6.02214076e23

def mean_count(c_nm, v_fl):
    return NA * 1e-24 * c_nm * v_fl

def empty_probability(mean):
    return exp(-mean)

def cv(mean):
    return 1 / sqrt(mean)

def mixing_ratio(c_nm, length_um=1, radius_um=0.004):
    return 4*pi*radius_um*length_um**2*mean_count(c_nm, 1)

def search_time_ratio(x):
    return (x + 1/x)/2

def occupancy(c, kd, hill=1):
    return c**hill / (c**hill + kd**hill)

def contrast(kd, hill=1):
    return occupancy(20, kd, hill) - occupancy(10, kd, hill)

def delivery_profile(x_um, length_um):
    return exp(-x_um/length_um)

def events_per_product(r):
    return 2*(1+r)

def turnover_randomness(r):
    return 1 - 1/(2*(1+r))  # only at S = KM in this mechanism

print("DNA fraction", pi*0.001**2*(4.6e6*0.00034))
print("mean count, empty, CV",
      mean_count(1, 1), empty_probability(mean_count(1, 1)),
      cv(mean_count(1, 1)))
print("presence and precision means", -log(0.01), 100)
for length in (0.5, 1, 2):
    print("mixing warning nM", length, 1/mixing_ratio(1, length))
print("patch association M^-1 s^-1", 4*50*0.001*NA*1e-15)
for kd in (1, sqrt(200), 100):
    print("Kd, contrast", kd, contrast(kd))
for kon in (2e5, 1e8):
    km, kcat = 1e-4, 10
    koff = km*kon - kcat
    r = koff/kcat
    print("kon, koff, mean wait, events, randomness",
          kon, koff, 2/kcat, events_per_product(r),
          turnover_randomness(r))

Sources and contributions

The eleven primary Lecture 2 contributions are linked at their points of use. The cross-lecture connection to Anqi Tu links through her attributed discussion in the Lecture 1 essay. Student links pin the accepted Week 1 snapshot at commit c419637, checked on 8 September 2026. The following full texts were inspected for the claims made here.

1. M. B. Elowitz, M. G. Surette, P.-E. Wolf, J. B. Stock and S. Leibler. “Protein mobility in the cytoplasm of Escherichia coli.” Journal of Bacteriology 181, 197–203 (1999). Protein- and context-specific mobility measurements. Published article.
2. H.-X. Zhou, G. Rivas and A. P. Minton. “Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences.” Annual Review of Biophysics 37, 375–397 (2008). Association-rate analysis, especially pp. 380–381. Published review.
3. H. C. Berg and E. M. Purcell. “Physics of chemoreception.” Biophysical Journal 20, 193–219 (1977). Small absorbing patch and finite receptor capture, pp. 194–196. Published article.
4. A. D. Riggs, S. Bourgeois and M. Cohn. “The lac repressor–operator interaction. III. Kinetic studies.” Journal of Molecular Biology 53, 401–417 (1970). The association constant on p. 401 is 7 × 10⁹ M⁻¹s⁻¹. Published article.
5. P. Hammar, P. Leroy, A. Mahmutovic, E. G. Marklund, O. G. Berg and J. Elf. “The lac repressor displays facilitated diffusion in living cells.” Science 336, 1595–1598 (2012). Sliding estimate and explicit imperfect-recognition qualification of the search optimum. Published article.
6. L. Bintu and colleagues. “Transcriptional regulation by the numbers: models.” Current Opinion in Genetics & Development 15, 116–124 (2005). Inspected the full 2004 preprint, including assumptions linking binding occupancy to expression. Published version · Inspected preprint.
7. B. M. Bordeau, Y. Yang and J. P. Balthasar. “Transient competitive inhibition bypasses the binding site barrier to improve tumor penetration of trastuzumab and enhance T-DM1 efficacy.” Cancer Research 81, 4145–4154 (2021). Methods, Figure 4 and Table 2 establish the imaging metric and experimental setting. Published article.
8. G. E. Briggs and J. B. S. Haldane. “A note on the kinetics of enzyme action.” Biochemical Journal 19, 338–339 (1925). The mean curve does not determine the microscopic dissociation-to-conversion ratio. Published article.
9. L. B. Sullivan, D. Y. Gui, A. M. Hosios, L. N. Bush, E. Freinkman and M. G. Vander Heiden. “Supporting aspartate biosynthesis is an essential function of respiration in proliferating cells.” Cell 162, 552–563 (2015). Introduction, Results and Figure 1 provide the cofactor-recycling and alternative-electron-acceptor context. Published article.