How to read a living landscape

Pack a microscope, a sampling bottle, and a notebook with room for new words. We are going to read the same living world in three ways.

Under the microscope, a sheet of cells has directions. In the river, fragments of DNA have journeys. In the notebook, specimens have categories. Each gives us a map, and each map has something particular it must preserve.

For the cells, local directions must fit together across space. For the DNA, a detected trace must be connected to a possible origin. For the notebook, a new distinction should still let us recover an old answer. Epistemic Physics invites us to keep those relationships in the state itself: how descriptions connect, where tracelets came from, and which question a probability answers.

Let’s begin with a compass that has two equally good ends.

A cell’s compass has no arrowhead

An elongated cell might point along a northeast-southwest axis. Turning that axis through ${180}^{\circ }$ gives the same axis. Label its angle $\theta$, measured counterclockwise from the horizontal. Then $\theta$ and $\theta +\pi$ describe the same orientation, because $\pi$ radians is half a turn.

This kind of head-tail symmetric alignment is called nematic order. It appears in liquid crystals, and it can also be useful for describing elongated cells. To convert an axis into an ordinary directed arrow, double the angle:

The bold symbol $\mathbf{q}$ names a two-component vector. Its first component points horizontally and its second vertically. Reversing a cell’s axis now turns this arrow through $2\pi$, one complete revolution, so the arrow returns to where it started.

The doubled-angle circle resembles other directional state spaces in Epistemic Physics, but it has a distinctive physical interpretation: it encodes an axis with interchangeable ends. That interpretation tells us which configurations count as equivalent.

Now walk around a small patch of tissue. At each step, follow the local cell axis smoothly into the next one. What happens when you return to your starting place?

The loop remembers how you went around

Imagine that the doubled-angle arrow turns once while you walk around the patch. The cell axis has turned only half as far. It comes home with the same physical orientation, even though a continuously followed angle has advanced by $\pi$.

The arrow’s winding number $w$ counts its complete revolutions along the loop. Positive means counterclockwise and negative means clockwise. The corresponding defect charge $s$ of the axis field is

The loop symbol $\oint$ means “add the small angle changes all the way around a closed path.” Axis charges can therefore be half-integers: $+1/2$, $-1/2$, $+1$, and so on.

A nonzero winding has a consequence. Try to fill the loop’s interior with a continuous, everywhere well-defined unit axis. Somewhere that construction must break down. The center may lose a definite orientation or contain a singular pattern. That obstruction is a topological defect. Topology studies features that persist through smooth deformations, such as winding around a hole.

This is already a living connection. In experiments on epithelial monolayers, sheets of cells one layer thick, Saw and colleagues connected defects in cell alignment with patterns of stress, cell death, and extrusion, where a cell leaves the layer. Their study makes tissue geometry a place to look for biological events.

Make a prediction: if you reverse just one cell by ${180}^{\circ }$, should the defect charge change? What if you disturb a cell slightly and let its neighbors smooth out the disturbance?

The laboratory joins neighboring doubled arrows by the shorter arc. If two neighboring cell axes differ by exactly ${90}^{\circ }$, their doubled arrows oppose each other. There are two equally short ways around the circle, and that sampled pair alone cannot choose between them. The “unresolved” state is an invitation to inspect what lies between the sampled cells.

Give agreement a local dynamics

Let’s choose a simple rule for the ring. Neighboring cells prefer parallel axes. Assign the mismatch cost

The index $i$ labels cells around the ring, with the last linked back to the first. This dimensionless cost is zero when all neighboring axes agree. It increases when they disagree.

Suppose each axis moves downhill in this cost, according to ${\dot{\theta }}_i=-\partial E/\partial {\theta }_i$. A dot means a derivative with respect to our chosen relaxation time. Differentiating gives

The rule uses only the two neighbors. Yet the result can retain a global winding. In the smooth $+1/2$ preset, each axis turns by $\pi /12$ between adjacent cells. The pulls from the two sides balance, so the pattern is a stationary state of this ring model. A small local disturbance can relax back toward it.

The geometry gives us a new sort of memory: a pattern distributed through relationships. There is no single cell that stores the loop’s complete winding. It belongs to how the cells fit together.

This suggests a larger experiment in development. Let a tissue grow by cell division. Each daughter inherits a lineage and an axis, while division changes the neighborhood graph. Can a local rule keep a useful global pattern intact as the graph grows? Could some defects guide where a tissue folds, closes, or replaces a cell? We could couple this spatial rule to the gene switches in A cell chooses what to remember and ask how cell identity and collective shape guide each other.

Now lift your eye from the microscope. Our next map is moving downstream.

A fragment in the bottle has a past

A water sample can contain DNA released by an organism somewhere else. Environmental DNA, often shortened to eDNA, is genetic material collected from an environment rather than directly from the organism we want to detect. Flowing water gives it a route.

Deiner and Altermatt sampled a river downstream of lake-dwelling invertebrates and detected transported DNA far from the lake source. Their river experiment supplies a wonderfully concrete question: when a bottle contains a matching fragment, where did its journey begin?

Choose a straight channel. Let $x$ be distance in kilometres, elapsed hours, $u$ the current speed in kilometres per hour, $D$ a mixing coefficient in square kilometres per hour, and $\kappa$ a decay rate per hour. A unit pulse released at position $x_0$ has the transport model

$C$ is the surviving pulse amount per kilometre. Its center travels to $x_0+ut$, its spatial variance grows to $2Dt$, and its total surviving amount is $e^{-\kappa t}$. These three quantities answer different questions: where the pulse went, how widely it mixed, and how much remained.

The pulse is the solution of an advection-diffusion-decay equation. Advection carries material with the current; diffusion spreads it; decay removes it:

We now have a mathematical dictionary for a tracelet’s age and route. A sample can be interpreted by asking which possible origin, passed through this transport rule, best predicts the observation.

Let the river change the odds

Suppose source A or source B released the pulse. Give the two hypotheses equal prior probability. Our sampling model predicts a mean captured count ${\mu }_A$ under A and ${\mu }_B$ under B. A Poisson count model gives the probability of observing $k$ fragments as

where $k$ is a nonnegative whole number and $k!$ is its factorial. Let $\ell$ name the log of the likelihood ratio. Dividing the likelihoods gives

Here is the EP bridge: half this log-likelihood ratio is the rapidity increment of the source odds. A physical travel model has become a transformation of a belief state. The two sources’ current probabilities come from both the observation and the transport assumptions used to read it.

Rapidity here means $\rho =\frac{1}{2}\log (p_A/p_B)$, where $p_A$ and $p_B$ are the probabilities of the two source hypotheses. Equal priors start at $\rho =0$. After this observation, $\rho =\ell /2$, and $p_A=1/(1+e^{-2\rho })$.

For $k=0$, the first term disappears. An empty bottle favors the hypothesis that predicted fewer captures. The sampling effort belongs in $\mu$; searching for one minute and searching all afternoon are different experiments.

Try this: keep the same hypothetical bottle and fragment count, but compare flowing water with still water. Can the likely source change?

The demo chooses $\mu =0.05+bC$, with background mean $0.05$ and a capture factor $b=8$ kilometres. Because $C$ is pulse amount per kilometre, this factor converts the unit-pulse density into a dimensionless expected count. It combines the release amount and sampling efficiency; it is a parameter of our experiment.

Moving the bottle marker edits the location of this one hypothetical sample. Collecting another bottle would require adding another observation with its own location, time, and source relationships. Multiple DNA fragments may share an organism or release event. The question determines which identity matters, just as a family tree changes the distinction between descendants and mutation origins in Evolution has an order.

There is a rich oceanographic extension here. Replace the straight current with a moving velocity field, allow traces to sink or attach to particles, and infer a distribution over possible origins and ages. A ship could choose its next station by asking which new sample would separate the remaining source histories most effectively. The field notebook becomes a map of possible journeys.

Give a trace a finite travel cone

The diffusion model spreads a pulse into tails extending throughout the line at every positive time. Let’s construct a different microscopic story with a finite speed.

Let fragments move either right or left at speed , switching direction at rate . This rate means that a short time $dt$ carries approximately probability $\gamma dt$ of a switch. Let $c_{+}(x,t)$ and $c_{-}(x,t)$ be their right-moving and left-moving densities. With decay rate $\kappa$,

Define total density $C=c_{+}+c_{-}$ and flux $J=v(c_{+}-c_{-})$. A flux is the amount crossing a position per unit time. Adding and subtracting the two equations yields

Remove decay by writing $U=e^{\kappa t}C$. Eliminating the corresponding flux gives the telegraph equation,

For a localized release, every path lies inside $|x-x_0|\le vt$. The resulting travel cone comes directly from the chosen velocity bound. Persistent random motion and the telegraph equation have a long mathematical connection, developed in work such as Goldstein’s study.

At times much longer than $1/(2\gamma )$, the flux can approximately follow the density gradient. Then $U$ approaches a diffusion description with $D=v^2/(2\gamma )$. A current adds the displacement $ut$ to the cone’s center.

We can now ask an ambitious question with explicit parts: what biological information is lost when a transport model forgets its directional persistence? The finite-speed model keeps a flux memory and a travel boundary. The diffusion approximation keeps a simpler long-time spread. An experiment with time-resolved release and capture could compare those predictions.

Our map has acquired a clock and a cone. It still needs names for what we find.

Discover a new category without rewriting the past

Your plankton notebook has three groups: diatoms, other known organisms, and a mystery group. Later, the stored images reveal two distinct mystery forms. You give them separate names.

What should happen to your estimate of the probability that a new specimen belongs to the original mystery group?

Use a Dirichlet distribution, a distribution over the probabilities of several mutually exclusive categories. It generalizes the Beta family. Category $i$ has prior weight and observed trial count $n_i$. After independent categorical specimen observations, its posterior weight is ${\alpha }_i+n_i$.

Take EP’s chosen unit-total directional prior, $\sum _i{\alpha }_i=1$. With three original groups, choose weight $1/3$ for each. When mystery group C splits into C₁ and C₂, divide its prior weight:

The new child weights might be $1/6$ each. A different split is possible if the prior model gives a reason for it. In either case the total prior remains one, and every saved specimen retains its identity.

Dirichlet distributions have an exact aggregation property: summing category probabilities gives another Dirichlet distribution with the corresponding weights summed. Write $a_i={\alpha }_i+n_i$ for each category’s posterior weight. The mystery group has combined weight $a_C$, and all the other groups have combined weight $b_C$:

The probability $p_C$ of belonging to either mystery child has the Beta marginal

A marginal is the distribution for a selected quantity after the other distinctions have been combined or integrated out. The parameters above equal those of the original coarse mystery-group model. Thus the whole coarse distribution survives, including its mean and variance. Ferguson’s treatment develops the Dirichlet properties behind such consistent partitions.

The saved images in this experiment already contain the distinguishing features. Revealing the labels teaches us which finer type each specimen belongs to, while adding no new specimen trial. Collecting another specimen is a different operation, and the demo lets you compare them.

This is useful far beyond naming plankton. A cell atlas can acquire new cell types. An immune model can refine the classes of encounters it distinguishes. An ecological survey can separate a broad group into species while still answering a long-running question about the original group. A finer question can inherit an older lineage of observations.

Let a living system help choose its maps

We have followed three kinds of structure. The tissue loop preserves a relation among local directions. The molecular sample preserves a connection to possible source histories. The changing taxonomy preserves coarse answers while opening finer questions.

Now imagine a model that can change all three.

A developing tissue contains cells with persistent internal states, local axes, and receptors tuned to nearby signals. Cells divide, so the lineage and the neighborhood graph grow together. Signals travel with their own times and routes. New cell fates introduce new categories, while broader distinctions remain available. Each local observer would carry a small description of the world that its current tasks can use.

Start with a ring of gene-switch cells exchanging a transported signal. Ask whether local receptor adaptation helps preserve its pattern when the environmental concentration scale changes. Then let cells divide, and ask how often the growth rule preserves winding. Finally split one broad fate into two subtypes and examine which coarse predictions survive the refinement.

At the tissue scale this could become a model of robust development. At the ecological scale it could become a network of organisms and sensors jointly discovering useful descriptions of their environment. At the most ambitious scale, life might be studied as matter that continually builds, maintains, and revises the maps by which it acts.

The microscope, the bottle, and the notebook have brought us to the same next experiment: let the descriptions change, and see which relationships a living world can keep.