Chapter map

1. 1. Spliced and unspliced inputs
2. 2. Transcription equation
3. 3. Splicing and degradation equation
4. 4. Four transcriptional states
5. 5. Gene-specific rates
6. 6. Low-dimensional cell state
7. 7. State posterior
8. 8. Latent-time decoder
9. 9. Dynamical solutions
10. 10. Variational objective
11. 11. Posterior velocity samples
12. 12. Intrinsic uncertainty
13. 13. Extrinsic uncertainty
14. 14. Worked one-gene derivative
15. 15. Permutation suitability test
16. 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.

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1. Spliced and unspliced inputs

Here the geometry becomes operational. Each cell–gene pair has measured or estimated unspliced u and spliced s abundances.

A concrete situation

A gene has u=4 and s=12 in one cell.

What the mathematics does not license

Quantification pipelines disagree on ambiguous reads. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Repeat preprocessing. 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. Transcription equation

This step deserves a careful pause. Unspliced RNA changes as du/dt = alpha – beta u during an active transcriptional state.

A concrete situation

Production alpha exceeds splicing loss beta u.

What the mathematics does not license

Constant alpha may be unrealistic. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Compare time-dependent rates. 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. Splicing and degradation equation

The useful question is not only how to calculate, but what the calculation assumes. Spliced RNA changes as ds/dt = beta u – gamma s.

A concrete situation

New spliced RNA enters at beta u and leaves at gamma s.

What the mathematics does not license

Rates can be weakly identified. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Inspect posterior correlation. 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. Four transcriptional states

A small example makes the mechanism visible. States represent induction, induction steady state, repression and repression steady state.

A concrete situation

A gene switches from induction to repression at t_s.

What the mathematics does not license

Real systems may branch or pulse. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use model criticism. 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. Gene-specific rates

This is where modelling choices begin to affect evidence. Alpha, beta and gamma vary by gene, with constraints that keep rates positive.

A concrete situation

Fast-splicing gene B has larger beta than gene A.

What the mathematics does not license

Gene-wise independence is an approximation. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Check coordinated 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.

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6. Low-dimensional cell state

The central mathematical idea is concrete. A ten-dimensional latent variable in the paper ties gene-specific times and states across the transcriptome.

A concrete situation

Cell z coordinates several simultaneous programmes.

What the mathematics does not license

Compression can blend lineages. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Vary latent width. 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. State posterior

Here the geometry becomes operational. An encoder outputs probabilities for induction and repression-related states for each cell–gene pair.

A concrete situation

A gene is 70% likely to be in repression.

What the mathematics does not license

Ambiguous states need uncertainty. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Avoid hard labels too early. 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. Latent-time decoder

This step deserves a careful pause. A neural decoder maps cell representation and state into gene-wise latent time.

A concrete situation

Two genes assign related but unequal times to one cell.

What the mathematics does not license

Latent time is model scale, not hours by default. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Anchor only with evidence. 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. Dynamical solutions

The useful question is not only how to calculate, but what the calculation assumes. Closed-form exponential solutions map rates, initial conditions and elapsed latent time to expected u and s.

A concrete situation

e^{-beta t} controls decay toward steady state.

What the mathematics does not license

Numerical extremes cause instability. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Test limiting cases. 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. Variational objective

A small example makes the mechanism visible. Likelihoods of u and s plus KL terms train encoders, rates and decoder jointly.

A concrete situation

One gradient update improves several genes.

What the mathematics does not license

Fit can favor abundant genes. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Report gene-stratified 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.

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11. Posterior velocity samples

This is where modelling choices begin to affect evidence. Repeated latent and state draws produce a distribution of beta u – gamma s for every cell and gene.

A concrete situation

Mean velocity is positive with an interval crossing zero.

What the mathematics does not license

The mean hides sign uncertainty. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Report sign probabilities. 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. Intrinsic uncertainty

The central mathematical idea is concrete. Variation across posterior velocity samples measures uncertainty inside the fitted model.

A concrete situation

A cell has a wide credible interval for one gene.

What the mathematics does not license

It excludes model misspecification. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Combine with sensitivity tests. 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. Extrinsic uncertainty

Here the geometry becomes operational. Posterior velocities propagated through a neighbour graph also reflect disagreement among similar cells.

A concrete situation

Candidate future states scatter across neighbours.

What the mathematics does not license

Neighbour definitions can dominate. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Vary 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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14. Worked one-gene derivative

This step deserves a careful pause. With beta=0.4, u=5, gamma=0.1 and s=12, compute ds/dt = 2 – 1.2 = 0.8.

A concrete situation

Positive velocity suggests increasing spliced abundance locally.

What the mathematics does not license

Units and uncertainty were simplified. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Sample alternative rates. 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. Permutation suitability test

The useful question is not only how to calculate, but what the calculation assumes. Shuffling or structured permutation can reveal partially observed or steady-state dynamics where velocity interpretation is weak.

A concrete situation

A dataset fails to outperform a null direction.

What the mathematics does not license

Null design matters. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Publish the permutation scheme. 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

A small example makes the mechanism visible. Version quantification, genes, equations, priors, rates, graph and uncertainty. This joins calculus, probability and biology.

A concrete situation

Students can start with exponential decay.

What the mathematics does not license

A velocity arrow is a conditional derivative. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Write a veloVI evidence 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 veloVI audit connects snapshot RNA counts to differential equations. For every gene, transcription produces unspliced RNA, splicing moves material into the spliced pool, and degradation removes spliced RNA. The model learns transcription, splicing and degradation rates, a latent time and an induction or repression state while sharing information through a low-dimensional cell representation.

Record spliced and unspliced quantification, gene filters, kinetic equations, rate constraints, state definitions, switch times, latent dimension, likelihood, priors, posterior samples, velocity graph and uncertainty summaries. The published model returns cell-by-gene posterior velocity samples, which lets uncertainty travel into downstream transition calculations.

Validate on simulations outside tuning, time-labelled or perturbation experiments, preprocessing alternatives, permutation tests, steady-state diagnostics and direction-of-time checks. RNA velocity is a model-based local derivative, not a video of a cell, a guaranteed lineage, or proof that one mature state becomes another.

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Frequently asked questions

What is RNA velocity?

RNA velocity is a model-based estimate of the local rate and direction of change in gene expression, commonly using unspliced and spliced RNA abundances.

What does veloVI add?

veloVI fits a Bayesian deep generative dynamical model, shares information across cells and genes, and returns posterior samples of cell-by-gene velocity so uncertainty can be measured.

Which kinetic rates are learned?

The model includes transcription rate alpha, splicing rate beta and degradation rate gamma, together with gene switching times and latent transcriptional states.

What is latent time?

It is an inferred position along a gene-specific induction or repression process. It is not clock time unless separately anchored to measured time.

How is velocity calculated?

For spliced RNA, the dynamical equation is ds/dt = beta u – gamma s. The fitted unspliced u, spliced s and rates determine the local derivative.

Does a velocity arrow prove a lineage?

No. Direction depends on model assumptions, data quality, gene kinetics, neighbourhood construction and uncertainty. Time-labelled or perturbation evidence is needed for strong lineage claims.

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Primary source and fact-check note

Adam Gayoso, Philipp Weiler, Mohammad Lotfollahi et al., “Deep generative modeling of transcriptional dynamics for RNA velocity analysis in single cells,” Nature Methods 21, 50–59 (2024), DOI 10.1038/s41592-023-01994-w. 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

• CellFM continuous expression modelling
• scPRINT denoising and networks
• Nicheformer spatial context
• 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

Quantification audit

Repeat spliced and unspliced counting with defensible pipelines and gene filters.

Equation audit

Write every rate, state, initial condition, switch time and unit before fitting.

Posterior sign test

Report how often velocity is positive, negative or crosses zero across samples.

Preprocessing stability

Repeat normalization, neighbourhood and feature choices.

Time validation

Compare direction with measured time, lineage tracing or controlled perturbations.

Suitability null

Use permutations and steady-state checks before drawing arrows.

Model extension

Compare constant and time-dependent transcription rates where data support them.

Final claim

Name counts, equations, rates, states and uncertainty. Velocity is a local model derivative.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Spliced and unspliced inputs

Start with this article's mechanism: Each cell–gene pair has measured or estimated unspliced u and spliced s abundances. Work through the concrete case: A gene has u=4 and s=12 in one cell. Make the boundary visible: Quantification pipelines disagree on ambiguous reads. Complete the evidence step: Repeat preprocessing. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 2: Transcription equation

Start with this article's mechanism: Unspliced RNA changes as du/dt = alpha – beta u during an active transcriptional state. Work through the concrete case: Production alpha exceeds splicing loss beta u. Make the boundary visible: Constant alpha may be unrealistic. Complete the evidence step: Compare time-dependent rates. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 3: Splicing and degradation equation

Start with this article's mechanism: Spliced RNA changes as ds/dt = beta u – gamma s. Work through the concrete case: New spliced RNA enters at beta u and leaves at gamma s. Make the boundary visible: Rates can be weakly identified. Complete the evidence step: Inspect posterior correlation. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 4: Four transcriptional states

Start with this article's mechanism: States represent induction, induction steady state, repression and repression steady state. Work through the concrete case: A gene switches from induction to repression at t_s. Make the boundary visible: Real systems may branch or pulse. Complete the evidence step: Use model criticism. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 5: Gene-specific rates

Start with this article's mechanism: Alpha, beta and gamma vary by gene, with constraints that keep rates positive. Work through the concrete case: Fast-splicing gene B has larger beta than gene A. Make the boundary visible: Gene-wise independence is an approximation. Complete the evidence step: Check coordinated residuals. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 6: Low-dimensional cell state

Start with this article's mechanism: A ten-dimensional latent variable in the paper ties gene-specific times and states across the transcriptome. Work through the concrete case: Cell z coordinates several simultaneous programmes. Make the boundary visible: Compression can blend lineages. Complete the evidence step: Vary latent width. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 7: State posterior

Start with this article's mechanism: An encoder outputs probabilities for induction and repression-related states for each cell–gene pair. Work through the concrete case: A gene is 70% likely to be in repression. Make the boundary visible: Ambiguous states need uncertainty. Complete the evidence step: Avoid hard labels too early. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 8: Latent-time decoder

Start with this article's mechanism: A neural decoder maps cell representation and state into gene-wise latent time. Work through the concrete case: Two genes assign related but unequal times to one cell. Make the boundary visible: Latent time is model scale, not hours by default. Complete the evidence step: Anchor only with evidence. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 9: Dynamical solutions

Start with this article's mechanism: Closed-form exponential solutions map rates, initial conditions and elapsed latent time to expected u and s. Work through the concrete case: e^{-beta t} controls decay toward steady state. Make the boundary visible: Numerical extremes cause instability. Complete the evidence step: Test limiting cases. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 10: Variational objective

Start with this article's mechanism: Likelihoods of u and s plus KL terms train encoders, rates and decoder jointly. Work through the concrete case: One gradient update improves several genes. Make the boundary visible: Fit can favor abundant genes. Complete the evidence step: Report gene-stratified residuals. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 11: Posterior velocity samples

Start with this article's mechanism: Repeated latent and state draws produce a distribution of beta u – gamma s for every cell and gene. Work through the concrete case: Mean velocity is positive with an interval crossing zero. Make the boundary visible: The mean hides sign uncertainty. Complete the evidence step: Report sign probabilities. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 12: Intrinsic uncertainty

Start with this article's mechanism: Variation across posterior velocity samples measures uncertainty inside the fitted model. Work through the concrete case: A cell has a wide credible interval for one gene. Make the boundary visible: It excludes model misspecification. Complete the evidence step: Combine with sensitivity tests. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 13: Extrinsic uncertainty

Start with this article's mechanism: Posterior velocities propagated through a neighbour graph also reflect disagreement among similar cells. Work through the concrete case: Candidate future states scatter across neighbours. Make the boundary visible: Neighbour definitions can dominate. Complete the evidence step: Vary graph parameters. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 14: Worked one-gene derivative

Start with this article's mechanism: With beta=0.4, u=5, gamma=0.1 and s=12, compute ds/dt = 2 – 1.2 = 0.8. Work through the concrete case: Positive velocity suggests increasing spliced abundance locally. Make the boundary visible: Units and uncertainty were simplified. Complete the evidence step: Sample alternative rates. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 15: Permutation suitability test

Start with this article's mechanism: Shuffling or structured permutation can reveal partially observed or steady-state dynamics where velocity interpretation is weak. Work through the concrete case: A dataset fails to outperform a null direction. Make the boundary visible: Null design matters. Complete the evidence step: Publish the permutation scheme. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 16: Governance and learning

Start with this article's mechanism: Version quantification, genes, equations, priors, rates, graph and uncertainty. This joins calculus, probability and biology. Work through the concrete case: Students can start with exponential decay. Make the boundary visible: A velocity arrow is a conditional derivative. Complete the evidence step: Write a veloVI evidence card. Write the count matrix, binary region table, differential equation or spatial mixture, then name every unit, probability assumption, latent variable and uncertainty source. Repeat under a nearby defensible setting and compare negative binomial tails, library factors, Bernoulli products, kinetic derivatives, posterior signs, cell-type proportions or residual maps as relevant. Use complete donors, studies, batches, time points or tissue sections outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.