Chapter map

1. 1. Molecular input
2. 2. Initial cell clustering
3. 3. Physical neighbourhood
4. 4. MECN definition
5. 5. Composition histogram
6. 6. Cell-cluster graph
7. 7. Ground distance
8. 8. Earth mover’s distance
9. 9. Optimal transport plan
10. 10. Pairwise MECN distances
11. 11. Microenvironment graph
12. 12. Spatial heterogeneity
13. 13. Spatial-domain clustering
14. 14. Differential microenvironment
15. 15. Three-dimensional support
16. 16. Governance and learning

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1. Molecular input

The central mathematical idea is concrete. Cells or spots carry expression, protein or metabolic features.

A concrete situation

A multiplexed tissue measures dozens of proteins.

What the mathematics does not license

Panel choice limits cell-state resolution. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit marker coverage. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Initial cell clustering

Here the geometry becomes operational. Molecular profiles are grouped into cell types or states.

A concrete situation

Cells receive one of six cluster labels.

What the mathematics does not license

Clustering errors enter every neighbourhood. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Repeat label resolutions. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Physical neighbourhood

This step deserves a careful pause. Coordinate nearest neighbours surround each centre cell.

A concrete situation

Thirty nearby cells form one local context.

What the mathematics does not license

Fixed k spans different radii in dense regions. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Plot physical reach. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. MECN definition

The useful question is not only how to calculate, but what the calculation assumes. The centre cell indexes one molecular-expression-aware cellular neighbourhood.

A concrete situation

Cell i and its neighbours form MECN i.

What the mathematics does not license

Overlapping MECNs are statistically dependent. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use spatially aware validation. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Composition histogram

A small example makes the mechanism visible. Counts or proportions across cell types represent the MECN.

A concrete situation

A vector is 0.5, 0.3 and 0.2.

What the mathematics does not license

Histograms discard exact neighbour arrangement. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare ring or directional features. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Cell-cluster graph

This is where modelling choices begin to affect evidence. Relationships among cell types support a biological ground geometry.

A concrete situation

Related neuronal types connect strongly.

What the mathematics does not license

The graph depends on embedding and clusters. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Test alternative constructions. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Ground distance

The central mathematical idea is concrete. A cost specifies how difficult it is to move mass between cell-type bins.

A concrete situation

Related types cost one; distant types cost three.

What the mathematics does not license

Ground costs encode assumptions. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Publish the matrix. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Earth mover’s distance

Here the geometry becomes operational. A transport plan minimizes total ground cost between histograms.

A concrete situation

Mass shifts from type A to B and C.

What the mathematics does not license

Normalization and unequal totals matter. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

State mass convention. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Optimal transport plan

This step deserves a careful pause. A nonnegative matrix satisfies source and target marginals.

A concrete situation

Each entry records abstract moved composition mass.

What the mathematics does not license

Plans may be non-unique. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report distance and ambiguity. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Pairwise MECN distances

The useful question is not only how to calculate, but what the calculation assumes. Every pair of neighbourhoods receives a composition distance.

A concrete situation

Similar boundaries have small EMD.

What the mathematics does not license

All-pairs computation can be costly. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Benchmark approximation. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Microenvironment graph

A small example makes the mechanism visible. Low-distance MECNs become connected graph nodes.

A concrete situation

Neighbourhoods with similar composition form a manifold.

What the mathematics does not license

Graph k changes connectivity. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Sweep graph parameters. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Spatial heterogeneity

This is where modelling choices begin to affect evidence. Neighbouring MECNs with different compositions create local heterogeneity.

A concrete situation

A cortex depth gradient raises a spatial score.

What the mathematics does not license

Edges and sampling can mimic heterogeneity. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use permutation controls. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Spatial-domain clustering

The central mathematical idea is concrete. Graph communities group related microenvironments.

A concrete situation

A tissue compartment contains similar MECNs.

What the mathematics does not license

A domain is a composition pattern, not one cell type. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Validate anatomical markers. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Differential microenvironment

Here the geometry becomes operational. Samples are compared by the relative occurrence of MECN states.

A concrete situation

One tumour enriches a boundary neighbourhood.

What the mathematics does not license

Sample size and batch confound abundance. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use donor-level replication. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Three-dimensional support

This step deserves a careful pause. Coordinates may be two- or three-dimensional when neighbourhoods are defined correctly.

A concrete situation

Cells in a volume include z distance.

What the mathematics does not license

Anisotropic z resolution distorts distance. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Rescale physical axes. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Governance and learning

The useful question is not only how to calculate, but what the calculation assumes. Version labels, neighbourhoods, ground costs, transport, graph and tests.

A concrete situation

A student solves a three-bin transport problem.

What the mathematics does not license

A microenvironment graph is a summary. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write a SOTIP audit card. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

SOTIP treats the microenvironment around each cell as the object to compare. It first assigns cell types or molecular clusters, searches the physical neighbours of every centre cell and records the neighbourhood as a histogram over those types. Each molecular-expression-aware cellular neighbourhood, or MECN, is therefore a composition rather than a single-cell vector.

Two histograms can contain the same number of cells yet differ in which types are present. SOTIP compares them with earth mover’s distance, an optimal-transport cost that moves histogram mass between cell-type bins using a biologically informed ground distance. These pairwise distances define a microenvironment graph on which clustering, spatial heterogeneity and differential microenvironment analyses can operate.

The answer depends on the first cell clustering, neighbourhood size, ground-distance construction, transport normalization and graph parameters. Earth mover’s distance describes the least-cost rearrangement between compositions; it does not identify actual cell movement or communication. A boundary can appear heterogeneous because of anatomy, sampling density or segmentation uncertainty.

A useful calculation compares two three-bin neighbourhood histograms. Give nearby cell types a transport cost of one and distant types a cost of three, then find the cheapest flow that converts one histogram into the other. Change the ground cost and observe the new microenvironment graph edge. The exercise shows why geometry inside the histogram space matters.

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Frequently asked questions

What is SOTIP?

SOTIP is a graph-based framework that represents the microenvironment around each cell and supports several spatial-omics analyses.

What is a MECN?

A molecular-expression-aware cellular neighbourhood is the physical neighbourhood centred on one cell, summarized as a histogram of cell types or molecular clusters.

Why use earth mover’s distance?

It measures the minimum cost of transforming one neighbourhood histogram into another when moves between cell-type bins have specified ground costs.

What is the microenvironment graph?

Each MECN is a node, and similarities derived from transport distances define edges between neighbourhoods.

What can the graph support?

SOTIP uses it for spatial heterogeneity quantification, spatial-domain identification and differential microenvironment analysis.

Does transport mass describe real cell motion?

No. It is an optimization device for comparing compositions; it does not track cells or prove communication.

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Primary source and fact-check note

Zhiyuan Yuan, Yisi Li, Minglei Shi et al., “SOTIP is a versatile method for microenvironment modeling with spatial omics data,” Nature Communications 13, 7330 (2022), DOI 10.1038/s41467-022-34867-5. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• MENDER multirange compositions
• BANKSY neighbourhood kernels
• SpaGCN domain graphs
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Label audit

Repeat cell-type clustering and marker checks.

Neighbour audit

Plot physical radii under fixed k.

Histogram audit

Compare counts, proportions and directional loss.

Ground audit

Publish cell-cluster costs and alternatives.

Transport audit

Check marginal mass, plan ambiguity and approximation.

Graph audit

Sweep MECN graph k and connectivity.

Differential audit

Use donor-level replication and batch controls.

Final claim

Transport geometry compares compositions; it does not track cells.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Molecular input

Begin with this mechanism: Cells or spots carry expression, protein or metabolic features. Work through the concrete case: A multiplexed tissue measures dozens of proteins. Make the limitation visible: Panel choice limits cell-state resolution. Complete the evidence step: Audit marker coverage. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 2: Initial cell clustering

Begin with this mechanism: Molecular profiles are grouped into cell types or states. Work through the concrete case: Cells receive one of six cluster labels. Make the limitation visible: Clustering errors enter every neighbourhood. Complete the evidence step: Repeat label resolutions. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 3: Physical neighbourhood

Begin with this mechanism: Coordinate nearest neighbours surround each centre cell. Work through the concrete case: Thirty nearby cells form one local context. Make the limitation visible: Fixed k spans different radii in dense regions. Complete the evidence step: Plot physical reach. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 4: MECN definition

Begin with this mechanism: The centre cell indexes one molecular-expression-aware cellular neighbourhood. Work through the concrete case: Cell i and its neighbours form MECN i. Make the limitation visible: Overlapping MECNs are statistically dependent. Complete the evidence step: Use spatially aware validation. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 5: Composition histogram

Begin with this mechanism: Counts or proportions across cell types represent the MECN. Work through the concrete case: A vector is 0.5, 0.3 and 0.2. Make the limitation visible: Histograms discard exact neighbour arrangement. Complete the evidence step: Compare ring or directional features. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 6: Cell-cluster graph

Begin with this mechanism: Relationships among cell types support a biological ground geometry. Work through the concrete case: Related neuronal types connect strongly. Make the limitation visible: The graph depends on embedding and clusters. Complete the evidence step: Test alternative constructions. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 7: Ground distance

Begin with this mechanism: A cost specifies how difficult it is to move mass between cell-type bins. Work through the concrete case: Related types cost one; distant types cost three. Make the limitation visible: Ground costs encode assumptions. Complete the evidence step: Publish the matrix. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 8: Earth mover’s distance

Begin with this mechanism: A transport plan minimizes total ground cost between histograms. Work through the concrete case: Mass shifts from type A to B and C. Make the limitation visible: Normalization and unequal totals matter. Complete the evidence step: State mass convention. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 9: Optimal transport plan

Begin with this mechanism: A nonnegative matrix satisfies source and target marginals. Work through the concrete case: Each entry records abstract moved composition mass. Make the limitation visible: Plans may be non-unique. Complete the evidence step: Report distance and ambiguity. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 10: Pairwise MECN distances

Begin with this mechanism: Every pair of neighbourhoods receives a composition distance. Work through the concrete case: Similar boundaries have small EMD. Make the limitation visible: All-pairs computation can be costly. Complete the evidence step: Benchmark approximation. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 11: Microenvironment graph

Begin with this mechanism: Low-distance MECNs become connected graph nodes. Work through the concrete case: Neighbourhoods with similar composition form a manifold. Make the limitation visible: Graph k changes connectivity. Complete the evidence step: Sweep graph parameters. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 12: Spatial heterogeneity

Begin with this mechanism: Neighbouring MECNs with different compositions create local heterogeneity. Work through the concrete case: A cortex depth gradient raises a spatial score. Make the limitation visible: Edges and sampling can mimic heterogeneity. Complete the evidence step: Use permutation controls. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 13: Spatial-domain clustering

Begin with this mechanism: Graph communities group related microenvironments. Work through the concrete case: A tissue compartment contains similar MECNs. Make the limitation visible: A domain is a composition pattern, not one cell type. Complete the evidence step: Validate anatomical markers. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 14: Differential microenvironment

Begin with this mechanism: Samples are compared by the relative occurrence of MECN states. Work through the concrete case: One tumour enriches a boundary neighbourhood. Make the limitation visible: Sample size and batch confound abundance. Complete the evidence step: Use donor-level replication. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 15: Three-dimensional support

Begin with this mechanism: Coordinates may be two- or three-dimensional when neighbourhoods are defined correctly. Work through the concrete case: Cells in a volume include z distance. Make the limitation visible: Anisotropic z resolution distorts distance. Complete the evidence step: Rescale physical axes. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 16: Governance and learning

Begin with this mechanism: Version labels, neighbourhoods, ground costs, transport, graph and tests. Work through the concrete case: A student solves a three-bin transport problem. Make the limitation visible: A microenvironment graph is a summary. Complete the evidence step: Write a SOTIP audit card. Write the dimensions, counts, coordinates, kernels, covariance matrices, mixture probabilities, priors, graph edges, transport plans, objectives or update rules with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete sections, donors, regions, genes, cell types or landmarks outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.