WHY MATHEMATICS?

Why Mathematics? | STitch3D for Multislice Tissue Maps: 3D Neighbour Graphs, Graph Attention, Single-Cell References and Cell-Type Deconvolution

How STitch3D turns aligned serial sections and a matched single-cell atlas into joint spatial representations, coherent 3D regions and estimated cell-type distributions.

Why mathematics is important becomes easier to see when a real decision depends on it. This guide follows the mechanism from definitions and dimensions to a worked calculation, validation, practical learning and responsible use. It keeps factual evidence separate from interpretation and treats every illustrative scenario as an example rather than a promise of results.

Choose a reading route

Pre-align serial sections in a common coordinate frame
Build a graph that connects locations across adjacent slices
Learn spatial representations with graph attention
Estimate cell-type mixtures and test reference dependence

Chapter map

1. Serial-section inputs
2. Physical z positions
3. Pre-alignment transform
4. Combined coordinate system
5. 3D neighbour graph
6. Single-cell expression profiles
7. Shared gene matrix
8. Attention score
9. Neighbour normalization
10. Graph aggregation
11. Slice-specific effects
12. Gene- and spot-specific terms
13. Latent representation
14. Spatial-domain clustering
15. Cell-type deconvolution
16. Governance and learning

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

1. Serial-section inputs

The useful question is not only how to calculate, but what the calculation assumes. Each section supplies coordinates and a spot-by-gene count matrix.

A concrete situation

Ten coronal sections cover an adult mouse brain.

What the mathematics does not license

Sections may differ in coverage and quality. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Audit each slice before pooling. 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.

2. Physical z positions

A small example makes the mechanism visible. Slice order and spacing supply the third coordinate.

A concrete situation

Adjacent sections are 50 micrometres apart.

What the mathematics does not license

Index numbers are not physical distance. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record section thickness and gaps. 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.

3. Pre-alignment transform

This is where modelling choices begin to affect evidence. ICP or PASTE places x-y coordinates into one frame.

A concrete situation

A rotated section moves toward its neighbour.

What the mathematics does not license

Registration error enters every later graph edge. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Save transforms and residuals. 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.

4. Combined coordinate system

The central mathematical idea is concrete. Aligned x, y and z values form one 3D location matrix.

A concrete situation

Each spot receives a three-coordinate vector.

What the mathematics does not license

Axes may use incompatible scales. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Standardize physical units. 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.

Did You Know? In this chapter, the most important number is useful only together with its construction rule. Changing scaling, sample membership or validation design can change the numerical answer without changing the arithmetic.

5. 3D neighbour graph

Here the geometry becomes operational. Edges connect spatially nearby spots within and across slices.

A concrete situation

A spot connects to peers above and below.

What the mathematics does not license

Large gaps can create false neighbours. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Inspect cross-slice edge lengths. 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.

6. Single-cell expression profiles

This step deserves a careful pause. Annotated reference cells are summarized by cell type.

A concrete situation

Excitatory-neuron profiles become reference columns.

What the mathematics does not license

Reference labels may be incomplete. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Test reference and label alternatives. 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.

7. Shared gene matrix

The useful question is not only how to calculate, but what the calculation assumes. Spatial and reference datasets retain compatible genes.

A concrete situation

Highly variable genes support learning.

What the mathematics does not license

Selection can omit rare-cell markers. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Reserve markers for 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.

8. Attention score

A small example makes the mechanism visible. A trainable function scores each graph neighbour.

A concrete situation

One anatomically consistent neighbour receives more weight.

What the mathematics does not license

Attention is not causal importance. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Check stability across seeds. 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.

9. Neighbour normalization

This is where modelling choices begin to affect evidence. Softmax converts scores into weights that sum to one.

A concrete situation

Three neighbours receive 0.6, 0.3 and 0.1.

What the mathematics does not license

Crowded graphs dilute evidence. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare degree-aware graphs. 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.

10. Graph aggregation

The central mathematical idea is concrete. Weighted neighbour messages update spot representations.

A concrete situation

Cross-slice evidence sharpens a weak layer boundary.

What the mathematics does not license

Misaligned edges spread errors. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Ablate cross-slice edges. 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.

11. Slice-specific effects

Here the geometry becomes operational. Parameters account for systematic slice variation.

A concrete situation

One section has lower capture efficiency.

What the mathematics does not license

Technical and biological slice change can be confused. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use replicates and holdouts. 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.

12. Gene- and spot-specific terms

This step deserves a careful pause. The model represents systematic variation by genes and locations.

A concrete situation

A highly expressed housekeeping gene has its own baseline.

What the mathematics does not license

Extra flexibility may absorb real signals. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect fitted components. 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.

13. Latent representation

The useful question is not only how to calculate, but what the calculation assumes. Low-dimensional spot vectors combine expression and 3D context.

A concrete situation

Nearby homologous regions cluster together.

What the mathematics does not license

Embedding geometry is model dependent. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Validate independent anatomy. 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.

14. Spatial-domain clustering

A small example makes the mechanism visible. Latent vectors support coherent 3D region discovery.

A concrete situation

A brain layer persists across sections.

What the mathematics does not license

Cluster count changes boundaries. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Publish stability over 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.

15. Cell-type deconvolution

This is where modelling choices begin to affect evidence. Reference profiles combine to explain each spot.

A concrete situation

A spot estimates 70 per cent neurons and 30 per cent glia.

What the mathematics does not license

Proportions inherit reference bias. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use marker and simulation checks. 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.

16. Governance and learning

The central mathematical idea is concrete. Version registration, z spacing, graphs, genes, references and optimization.

A concrete situation

A student builds a two-slice adjacency matrix.

What the mathematics does not license

A reconstructed atlas is a model. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Write a 3D evidence ledger. 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.

A seven-day practice lab

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

STitch3D first places multiple approximately parallel tissue sections into a shared three-dimensional coordinate system, using ICP or PASTE for pre-alignment. It then constructs a 3D neighbour graph so a spot can borrow evidence within its slice and from nearby spots on adjacent slices.

A graph-attention model jointly uses spatial-transcriptomics counts, the 3D graph and cell-type-specific expression profiles derived from a matched single-cell RNA-sequencing reference. The model learns spot representations for spatial-domain discovery and estimates cell-type proportions across the reconstructed volume.

The model separates gene-, spot- and slice-specific contributions so shared biology can be strengthened while technical slice effects are controlled. That decomposition is an assumption, not a guarantee: a mismatched reference, incorrect pre-alignment, uneven slice spacing or missing tissue can create confident but wrong 3D patterns.

Frequently asked questions

What is STitch3D?

STitch3D jointly models multiple spatial-transcriptomics slices and a paired single-cell RNA-sequencing reference to reconstruct 3D regions and cell-type distributions.

Why is pre-alignment required?

A 3D graph needs comparable coordinates; STitch3D workflows use methods such as ICP or PASTE before graph construction.

What does graph attention do?

It learns data-dependent weights for neighbouring spots so their information contributes unequally to each updated representation.

How is the single-cell reference used?

Cell-type-specific expression profiles from annotated single-cell data guide deconvolution and representation learning.

What is a cell-type proportion?

It is an estimated nonnegative mixture for a spot under the model and reference, not a direct count of intact cells.

What can make a 3D pattern unreliable?

Misregistration, wrong slice spacing, missing tissue, batch effects or an unrepresentative reference can all create systematic error.

Primary source and fact-check note

Gefei Wang, Jia Zhao, Yan Yan, Yang Wang, Angela Ruohao Wu and Can Yang, “Construction of a 3D whole organism spatial atlas by joint modelling of multiple slices with deep neural networks,” Nature Machine Intelligence 5, 1200–1213 (2023), DOI 10.1038/s42256-023-00734-1. 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

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

Slice audit

Check coverage, counts, orientation and physical z positions.

Registration audit

Save pre-alignment transforms and residuals.

Graph audit

Inspect within- and cross-slice edge lengths.

Reference audit

Compare matched single-cell atlases and label sets.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Attention audit

Test edge weights and seed stability.

Deconvolution audit

Use simulations and withheld marker genes.

Domain audit

Repeat clustering resolutions and slice holdouts.

Final claim

A 3D atlas is reconstructed evidence, not direct observation.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Serial-section inputs

Begin with this mechanism: Each section supplies coordinates and a spot-by-gene count matrix. Work through the concrete case: Ten coronal sections cover an adult mouse brain. Make the limitation visible: Sections may differ in coverage and quality. Complete the evidence step: Audit each slice before pooling. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Physical z positions

Begin with this mechanism: Slice order and spacing supply the third coordinate. Work through the concrete case: Adjacent sections are 50 micrometres apart. Make the limitation visible: Index numbers are not physical distance. Complete the evidence step: Record section thickness and gaps. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Pre-alignment transform

Begin with this mechanism: ICP or PASTE places x-y coordinates into one frame. Work through the concrete case: A rotated section moves toward its neighbour. Make the limitation visible: Registration error enters every later graph edge. Complete the evidence step: Save transforms and residuals. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Combined coordinate system

Begin with this mechanism: Aligned x, y and z values form one 3D location matrix. Work through the concrete case: Each spot receives a three-coordinate vector. Make the limitation visible: Axes may use incompatible scales. Complete the evidence step: Standardize physical units. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 5: 3D neighbour graph

Begin with this mechanism: Edges connect spatially nearby spots within and across slices. Work through the concrete case: A spot connects to peers above and below. Make the limitation visible: Large gaps can create false neighbours. Complete the evidence step: Inspect cross-slice edge lengths. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Single-cell expression profiles

Begin with this mechanism: Annotated reference cells are summarized by cell type. Work through the concrete case: Excitatory-neuron profiles become reference columns. Make the limitation visible: Reference labels may be incomplete. Complete the evidence step: Test reference and label alternatives. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Shared gene matrix

Begin with this mechanism: Spatial and reference datasets retain compatible genes. Work through the concrete case: Highly variable genes support learning. Make the limitation visible: Selection can omit rare-cell markers. Complete the evidence step: Reserve markers for validation. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Attention score

Begin with this mechanism: A trainable function scores each graph neighbour. Work through the concrete case: One anatomically consistent neighbour receives more weight. Make the limitation visible: Attention is not causal importance. Complete the evidence step: Check stability across seeds. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 9: Neighbour normalization

Begin with this mechanism: Softmax converts scores into weights that sum to one. Work through the concrete case: Three neighbours receive 0.6, 0.3 and 0.1. Make the limitation visible: Crowded graphs dilute evidence. Complete the evidence step: Compare degree-aware graphs. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Graph aggregation

Begin with this mechanism: Weighted neighbour messages update spot representations. Work through the concrete case: Cross-slice evidence sharpens a weak layer boundary. Make the limitation visible: Misaligned edges spread errors. Complete the evidence step: Ablate cross-slice edges. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Slice-specific effects

Begin with this mechanism: Parameters account for systematic slice variation. Work through the concrete case: One section has lower capture efficiency. Make the limitation visible: Technical and biological slice change can be confused. Complete the evidence step: Use replicates and holdouts. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Gene- and spot-specific terms

Begin with this mechanism: The model represents systematic variation by genes and locations. Work through the concrete case: A highly expressed housekeeping gene has its own baseline. Make the limitation visible: Extra flexibility may absorb real signals. Complete the evidence step: Inspect fitted components. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 13: Latent representation

Begin with this mechanism: Low-dimensional spot vectors combine expression and 3D context. Work through the concrete case: Nearby homologous regions cluster together. Make the limitation visible: Embedding geometry is model dependent. Complete the evidence step: Validate independent anatomy. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Spatial-domain clustering

Begin with this mechanism: Latent vectors support coherent 3D region discovery. Work through the concrete case: A brain layer persists across sections. Make the limitation visible: Cluster count changes boundaries. Complete the evidence step: Publish stability over resolutions. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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: Cell-type deconvolution

Begin with this mechanism: Reference profiles combine to explain each spot. Work through the concrete case: A spot estimates 70 per cent neurons and 30 per cent glia. Make the limitation visible: Proportions inherit reference bias. Complete the evidence step: Use marker and simulation checks. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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 registration, z spacing, graphs, genes, references and optimization. Work through the concrete case: A student builds a two-slice adjacency matrix. Make the limitation visible: A reconstructed atlas is a model. Complete the evidence step: Write a 3D evidence ledger. Write the coordinates, graphs, matrices, costs, weights, transformations, losses, couplings or displacement fields with named dimensions and physical units. Calculate one small example by hand, then vary one assumption and explain which output moves and why. Reserve complete sections, donors, platforms, regions, genes 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.