Chapter map

1. 1. Why perturbation response is a counterfactual
2. 2. The training table
3. 3. Variational autoencoder
4. 4. Reparameterised sampling
5. 5. Control and perturbed centroids
6. 6. The perturbation delta
7. 7. Latent vector arithmetic
8. 8. Decoding to gene expression
9. 9. Out-of-sample cell types
10. 10. Cross-study and cross-species transfer
11. 11. Heterogeneous responses
12. 12. Nonlinearity and dose
13. 13. Gene-level validation
14. 14. Distribution-level validation
15. 15. A student vector project
16. 16. Governance and careers

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1. Why perturbation response is a counterfactual

The central mathematical idea is concrete. We observe a cell under control or treatment, not both. Prediction estimates the missing outcome.

A concrete situation

A control T cell’s stimulated transcriptome is counterfactual.

What the mathematics does not license

No model removes the need for experimental evidence. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Define treatment, target context and outcome. 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. The training table

Here the geometry becomes operational. Data need condition, cell type, donor and study labels with comparable genes. Splits must reflect the intended transfer.

A concrete situation

Train on several cell types and hold one out.

What the mathematics does not license

Random cells do not test unseen-context transfer. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Create a context-by-condition availability 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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3. Variational autoencoder

This step deserves a careful pause. The encoder maps expression to a posterior over latent z; the decoder maps latent samples back to expression.

A concrete situation

Similar cells can occupy nearby latent regions.

What the mathematics does not license

The latent space is model-dependent. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Draw the encoder, prior, sample and decoder. 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. Reparameterised sampling

The useful question is not only how to calculate, but what the calculation assumes. Write z as mu plus sigma times epsilon to train through stochastic samples.

A concrete situation

Posterior variance represents uncertainty in encoding.

What the mathematics does not license

Approximate inference can be miscalibrated. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Calculate three samples from toy 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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5. Control and perturbed centroids

A small example makes the mechanism visible. Average latent positions for control and perturbed training cells under a declared subset.

A concrete situation

Means summarise a population response.

What the mathematics does not license

A mean hides multimodal or rare responses. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compute weighted and unweighted centroids. 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. The perturbation delta

This is where modelling choices begin to affect evidence. Subtract the control centroid from the perturbed centroid to obtain a latent direction and magnitude.

A concrete situation

Delta represents an average response vector.

What the mathematics does not license

Confounded donors or composition can shape delta. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Calculate a two-dimensional delta. 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. Latent vector arithmetic

The central mathematical idea is concrete. Add delta to a target control cell or centroid to create a predicted perturbed latent state.

A concrete situation

z_pred=z_control+delta.

What the mathematics does not license

Additivity may fail across cell types or doses. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Apply one delta to three target cells. 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. Decoding to gene expression

Here the geometry becomes operational. The decoder turns predicted latent points into expression estimates or samples.

A concrete situation

Predicted means can be compared with observed perturbed cells.

What the mathematics does not license

Decoder smoothness can hide uncertainty. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report samples and intervals, not only means. 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. Out-of-sample cell types

This step deserves a careful pause. A key test withholds the target cell type’s perturbed cells during fitting and delta estimation.

A concrete situation

Predict stimulation in a cell type unseen under treatment.

What the mathematics does not license

Control cells from the target can still provide context. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Document exactly what was held out. 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. Cross-study and cross-species transfer

The useful question is not only how to calculate, but what the calculation assumes. The original work examined transfer across datasets and species under matched genes and response structure.

A concrete situation

Conserved response directions may generalise.

What the mathematics does not license

Orthology and platform shifts add assumptions. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Validate feature mapping and donor effects. 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. Heterogeneous responses

A small example makes the mechanism visible. Cells of one type may have responder and non-responder subgroups. A single mean vector can blur them.

A concrete situation

Treatment can change composition and covariance.

What the mathematics does not license

Average accuracy hides distributional failure. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare full predicted and observed distributions. 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. Nonlinearity and dose

This is where modelling choices begin to affect evidence. Real responses can saturate, reverse or interact with dose and time. One additive delta is a local approximation.

A concrete situation

Low and high dose can follow different paths.

What the mathematics does not license

Extrapolation beyond training is risky. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Plot response along dose and time if measured. 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. Gene-level validation

The central mathematical idea is concrete. Compare predicted and observed fold changes, directions, markers and pathways on held-out data.

A concrete situation

Known interferon genes provide targeted checks.

What the mathematics does not license

Correlation can be driven by unchanged genes. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report effect errors for responsive genes. 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. Distribution-level validation

Here the geometry becomes operational. Assess means, variances, covariance, subpopulation frequencies and calibrated intervals.

A concrete situation

Correct mean with wrong variance is incomplete.

What the mathematics does not license

A smooth decoder can understate tails. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use energy distance or other justified metrics. 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. A student vector project

This step deserves a careful pause. Simulate two cell types and one additive perturbation, then introduce an interaction that breaks transfer.

A concrete situation

Known truth exposes the additive boundary.

What the mathematics does not license

Real data add thousands of genes and donors. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Compare mean-vector and cell-specific models. 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 careers

The useful question is not only how to calculate, but what the calculation assumes. Version splits, genes, architecture, delta definition, target cells, seeds and validation. Perturbation prediction joins probability, vector geometry and experimental design.

A concrete situation

Students can begin with two-dimensional arrows.

What the mathematics does not license

Prediction guarantees no treatment effect or safety. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write a counterfactual model 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

A scGen audit starts with control and perturbed single-cell expression, cell-type labels, donors, conditions and a clearly defined prediction task. A variational autoencoder learns a latent representation, and an average perturbation difference vector is estimated by subtracting control from perturbed latent means.

Record training and held-out cell types, genes, normalisation, encoder and decoder, latent dimension, seeds, donor splits, cells used to estimate the delta vector and decoding procedure. Latent arithmetic assumes that a transferable response direction is meaningful for the target context.

Validate on cell types, donors, studies or species excluded from fitting and model selection. Compare predicted and observed distributions, gene-level effects, covariance and baselines. A useful counterfactual prediction remains a model-based forecast, not experimental evidence or a claim that every cell shares one additive response.

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Frequently asked questions

What does scGen predict?

It predicts single-cell expression responses to a perturbation for a target context, such as a held-out cell type.

What is a latent perturbation vector?

It is a difference between average perturbed and control latent representations under a defined training subset.

How is the prediction made?

The perturbation vector is added to a target control cell’s latent representation and decoded back to expression space.

Does one vector fit every cell type?

That is an assumption to test. Responses can be context-specific, nonlinear or heterogeneous.

Is a predicted response experimental evidence?

No. It is a counterfactual model prediction that requires held-out and experimental validation.

How should scGen be validated?

Hold out cell types, donors or studies; compare predicted and observed distributions, gene effects, covariance, subpopulations and simple baselines.

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Primary source and fact-check note

Mohammad Lotfollahi, F. Alexander Wolf and Fabian J. Theis, “scGen predicts single-cell perturbation responses,” Nature Methods 16, 715–721 (2019), DOI 10.1038/s41592-019-0494-8. 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

• scVI deep generative modelling
• DESC deep embedding clustering
• scANVI annotation uncertainty

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

Counterfactual estimand

Write the target quantity before modelling: the expression distribution the target cells would have shown under a named perturbation, dose and time. Specify donors, cell state and measurement platform. This prevents a broad phrase such as treatment response from hiding several different questions. The prediction remains counterfactual because the same biological cell cannot be observed under both outcomes.

Delta provenance

Store exactly which control and perturbed cells produced the latent difference vector, how cells and donors were weighted, and whether means were estimated within type before pooling. A delta dominated by one donor or a changing cell composition may encode confounding. Re-estimate it under donor leave-one-out analysis and report direction and magnitude stability.

Posterior-sample arithmetic

Apply the perturbation vector to posterior samples as well as posterior means. Decode many draws and report intervals for genes and summary scores. If uncertainty from encoding and decoding is ignored, a smooth mean profile can look more precise than the model supports. Sample-based checks also reveal whether vector addition moves cells into poorly supported latent regions.

Additivity stress test

Simulate responses that are purely additive, scaled by cell type and strongly nonlinear. Fit the same workflow and identify the point where one delta stops transferring. On real data, compare cell-type-specific observed deltas with the pooled vector. A transferable average direction is an empirical result, not a property guaranteed by vector notation.

Distributional validation

Compare predicted and observed means, variances, covariance, tails and subgroup frequencies on untouched perturbation data. A high correlation across thousands of mostly unchanged genes can coexist with large errors among responsive genes. Report effect-size error and direction for a prespecified responsive set, alongside whole-distribution metrics and simple baselines.

Dose-and-time boundary

A vector estimated at one dose and time should not be assumed valid elsewhere. Where measurements exist, plot latent displacement against dose and time and test whether direction or magnitude changes. Saturation, delay, reversal and interaction can all break linear transfer. Restrict the claim to the measured range unless external evidence supports extrapolation.

Experimental follow-up

Use predictions to prioritise experiments, not replace them. Select examples spanning high confidence, high uncertainty and biologically surprising effects, then measure the target context under perturbation. Pre-register evaluation genes and distributions. A model is most useful when it turns an enormous experimental space into a smaller, transparent set of falsifiable tests.

Decision boundary

Do not turn a predicted transcriptional response into a statement about treatment benefit, toxicity or clinical safety. Those outcomes require distinct evidence. The defensible conclusion names the dataset, held-out context, error, uncertainty and failure zones. That narrower sentence gives future researchers a reproducible claim they can challenge and improve.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Why perturbation response is a counterfactual

Start with the mechanism: We observe a cell under control or treatment, not both. Prediction estimates the missing outcome. The worked case is deliberately concrete: A control T cell’s stimulated transcriptome is counterfactual. Now expose the boundary: No model removes the need for experimental evidence. A student should complete this evidence step: Define treatment, target context and outcome. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 2: The training table

Start with the mechanism: Data need condition, cell type, donor and study labels with comparable genes. Splits must reflect the intended transfer. The worked case is deliberately concrete: Train on several cell types and hold one out. Now expose the boundary: Random cells do not test unseen-context transfer. A student should complete this evidence step: Create a context-by-condition availability matrix. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 3: Variational autoencoder

Start with the mechanism: The encoder maps expression to a posterior over latent z; the decoder maps latent samples back to expression. The worked case is deliberately concrete: Similar cells can occupy nearby latent regions. Now expose the boundary: The latent space is model-dependent. A student should complete this evidence step: Draw the encoder, prior, sample and decoder. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 4: Reparameterised sampling

Start with the mechanism: Write z as mu plus sigma times epsilon to train through stochastic samples. The worked case is deliberately concrete: Posterior variance represents uncertainty in encoding. Now expose the boundary: Approximate inference can be miscalibrated. A student should complete this evidence step: Calculate three samples from toy parameters. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 5: Control and perturbed centroids

Start with the mechanism: Average latent positions for control and perturbed training cells under a declared subset. The worked case is deliberately concrete: Means summarise a population response. Now expose the boundary: A mean hides multimodal or rare responses. A student should complete this evidence step: Compute weighted and unweighted centroids. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 6: The perturbation delta

Start with the mechanism: Subtract the control centroid from the perturbed centroid to obtain a latent direction and magnitude. The worked case is deliberately concrete: Delta represents an average response vector. Now expose the boundary: Confounded donors or composition can shape delta. A student should complete this evidence step: Calculate a two-dimensional delta. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 7: Latent vector arithmetic

Start with the mechanism: Add delta to a target control cell or centroid to create a predicted perturbed latent state. The worked case is deliberately concrete: z_pred=z_control+delta. Now expose the boundary: Additivity may fail across cell types or doses. A student should complete this evidence step: Apply one delta to three target cells. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 8: Decoding to gene expression

Start with the mechanism: The decoder turns predicted latent points into expression estimates or samples. The worked case is deliberately concrete: Predicted means can be compared with observed perturbed cells. Now expose the boundary: Decoder smoothness can hide uncertainty. A student should complete this evidence step: Report samples and intervals, not only means. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 9: Out-of-sample cell types

Start with the mechanism: A key test withholds the target cell type’s perturbed cells during fitting and delta estimation. The worked case is deliberately concrete: Predict stimulation in a cell type unseen under treatment. Now expose the boundary: Control cells from the target can still provide context. A student should complete this evidence step: Document exactly what was held out. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 10: Cross-study and cross-species transfer

Start with the mechanism: The original work examined transfer across datasets and species under matched genes and response structure. The worked case is deliberately concrete: Conserved response directions may generalise. Now expose the boundary: Orthology and platform shifts add assumptions. A student should complete this evidence step: Validate feature mapping and donor effects. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 11: Heterogeneous responses

Start with the mechanism: Cells of one type may have responder and non-responder subgroups. A single mean vector can blur them. The worked case is deliberately concrete: Treatment can change composition and covariance. Now expose the boundary: Average accuracy hides distributional failure. A student should complete this evidence step: Compare full predicted and observed distributions. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 12: Nonlinearity and dose

Start with the mechanism: Real responses can saturate, reverse or interact with dose and time. One additive delta is a local approximation. The worked case is deliberately concrete: Low and high dose can follow different paths. Now expose the boundary: Extrapolation beyond training is risky. A student should complete this evidence step: Plot response along dose and time if measured. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 13: Gene-level validation

Start with the mechanism: Compare predicted and observed fold changes, directions, markers and pathways on held-out data. The worked case is deliberately concrete: Known interferon genes provide targeted checks. Now expose the boundary: Correlation can be driven by unchanged genes. A student should complete this evidence step: Report effect errors for responsive genes. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 14: Distribution-level validation

Start with the mechanism: Assess means, variances, covariance, subpopulation frequencies and calibrated intervals. The worked case is deliberately concrete: Correct mean with wrong variance is incomplete. Now expose the boundary: A smooth decoder can understate tails. A student should complete this evidence step: Use energy distance or other justified metrics. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 15: A student vector project

Start with the mechanism: Simulate two cell types and one additive perturbation, then introduce an interaction that breaks transfer. The worked case is deliberately concrete: Known truth exposes the additive boundary. Now expose the boundary: Real data add thousands of genes and donors. A student should complete this evidence step: Compare mean-vector and cell-specific models. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.

Workshop 16: Governance and careers

Start with the mechanism: Version splits, genes, architecture, delta definition, target cells, seeds and validation. Perturbation prediction joins probability, vector geometry and experimental design. The worked case is deliberately concrete: Students can begin with two-dimensional arrows. Now expose the boundary: Prediction guarantees no treatment effect or safety. A student should complete this evidence step: Write a counterfactual model card. Record the inputs, mathematical operation, expected output, uncertainty and one result that would contradict the claim. Then repeat the calculation or model under a nearby defensible setting and compare what changes. This turns a polished output into an auditable argument. Use untouched donors, studies or simulated truth where possible, because random-cell checks can share hidden structure with training data. Keep measurement, model estimate, interpretation and decision as separate lines in the notebook.