Basic math LEGO bricks that I have used while building at the intersection of computers x biology. I use this as a reference.

Calculus + Linear Algebra

Component	Definition	In Biology / Computing	Math Variables
Derivative	Instantaneous rate of change	Enzyme kinetics, population growth, gradient descent	$\frac{dy}{dx}, \nabla f(x)$
Partial Derivative	Rate of change w.r.t. one variable	Multi-omics sensitivity, energy landscapes	$\frac{\partial f}{\partial x_i}$
Gradient	Vector of partial derivatives	Backpropagation, optimization	$\nabla f(x) = \left[\frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n}\right]$
Jacobian	Matrix of first-order derivatives	Sensitivity analysis, neural nets	$J_{ij} = \frac{\partial f_i}{\partial x_j}$
Hessian	Matrix of second-order derivatives	Curvature in protein folding, optimization	$H_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}$
Taylor Expansion	Approximation around a point	Local linearization of pathways, dynamics	$f(x) \approx f(a) + f'(a)(x-a) + \tfrac{1}{2}f''(a)(x-a)^2$
Chain Rule	Derivative of composite functions	Backpropagation in neural nets	$\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$
Integral	Accumulated quantity	Population size from growth rate, cumulative flux	$\int f(x)\, dx$
Definite Integral	Area under curve / accumulation over interval	Protein abundance from expression rate	$\int_a^b f(x)\, dx$
Multiple Integral	Integration over multidimensional space	Partition functions, probability densities	$\int \cdots \int f(x_1,\ldots,x_n) \, dx_1 \cdots dx_n$
Divergence	Outflow rate of a vector field	Flux of ions, transport phenomena	$\nabla \cdot \vec{F}$
Curl	Rotation of a vector field	Electromagnetic models in biophysics	$\nabla \times \vec{F}$
Laplacian	Divergence of gradient	Diffusion models, electrophysiology	$\nabla^2 f = \nabla \cdot (\nabla f)$
Linear Equation	Equation in vector/matrix form	Kinetics, statistical models	$Ax = b$
Matrix Multiplication	Transformation, composition	Feature embeddings, transition systems	$C = AB$
Determinant	Scalar property of matrix	Volume scaling, invertibility	$\det(A)$
Inverse Matrix	Solves linear systems	Regression, circuit models	$A^{-1}$
Eigenvalue / Eigenvector	Invariant scaling directions	PCA, network stability	$Av = \lambda v$
SVD	Decompose into orthogonal bases	Dimensionality reduction (scRNA-seq)	$A = U \Sigma V^T$
QR / LU / Cholesky	Matrix factorizations for solving	Numerical solvers for pathway models	$A = QR, \ A=LU, \ A=LL^T$
Projection	Mapping onto subspace	Embeddings, latent space representations	$P = UU^T$
Inner Product	Measure of similarity	Cosine similarity, kernel methods	$\langle x, y \rangle = x^T y$
Norm	Length / magnitude of vector	Regularization, error measures	$\\|x\\|p=\left(\sum{i=1}^{n}\|x_i\|^{\,p}\right)^{1/p}$
Mahalanobis Distance	Distance scaled by covariance	Anomaly detection in cell states	$d(x,y) = \sqrt{(x-y)^T \Sigma^{-1} (x-y)}$
Orthogonality	Perpendicular vectors/bases	PCA axes, Fourier modes	$x^T y = 0$
Rank	Dimension of column/row space	Degrees of freedom in system	$\text{rank}(A)$
Trace	Sum of diagonal elements	Invariant measure in statistics	$\text{tr}(A) = \sum_i A_{ii}$
Condition Number	Sensitivity of linear system	Numerical stability in simulations	$\kappa(A) = \\|A\\| \\|A^{-1}\\|$

Probability, Statistics, Information Theory

Component	Definition	In Biology / Computing	Math Variables
Probability Distribution	Assigns likelihood to outcomes	Gene expression variability, sequencing errors	$P(X=x)$
Expectation	Average value under distribution	Mean expression, fitness landscape averages	$\mathbb{E}[X] = \sum_x x P(x)$
Variance	Spread around the mean	Expression noise, measurement error	$\mathrm{Var}(X) = \mathbb{E}[(X-\mu)^2]$
Covariance	Joint variability of two variables	Co-expression of genes	$\mathrm{Cov}(X,Y) = \mathbb{E}[(X-\mu_X)(Y-\mu_Y)]$
Correlation	Normalized covariance (−1 to 1)	Gene-gene correlation networks	$\rho_{XY} = \frac{\mathrm{Cov}(X,Y)}{\sigma_X\sigma_Y}$
Law of Large Numbers	Averages converge to expectation	Replicates reduce noise	$\bar X_n=\frac{1}{n}\sum_{i=1}^{n}X_i \xrightarrow{\text{a.s.}} \mu \quad \text{as } n\to\infty$
Central Limit Theorem	Normal distribution emerges from sums	Sequencing depth → Gaussian errors	$\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \to \mathcal{N}(0,1)$
Bernoulli	Single binary outcome	Mutation present/absent	$P(X=1)=p, \ P(X=0)=1-p$
Binomial	Sum of Bernoullis	Count of mutations in reads	$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$
Poisson	Events in fixed interval	RNA counts, sequencing reads	$P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$
Negative Binomial	Overdispersed counts	scRNA-seq models	$P(X=k) = \binom{k+r-1}{k} (1-p)^r p^k$
Normal Distribution	Gaussian variability	Expression levels, errors	$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
Log-normal	Multiplicative noise	Protein abundance distributions	$f(x) = \frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}}$
Gamma	Waiting times, skewed distributions	Reaction times, lifetimes	$f(x;\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}$
Beta	Distribution on [0,1]	Allele frequencies, probabilities	$f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$
Dirichlet	Multivariate generalization of Beta	Topic/cell-type mixtures	$f{x};\alpha) = \frac{1}{B(\alpha)} \prod_i x_i^{\alpha_i - 1}$
Multinomial	Multi-class counts	Reads across cell types	$P(X=x_1,\ldots,x_k) = \frac{n!}{\prod_i x_i!} \prod_i p_i^{x_i}$
Maximum Likelihood (MLE)	Best-fit parameters from data	Fit kinetic rates, emission probs	$\hat{\theta}{\mathrm{MLE}}=\arg\max{\theta}\;L(\theta\mid x) =\arg\max_{\theta}\;\sum_{i=1}^{n}\log p(x_i\mid\theta)$
Maximum A Posteriori (MAP)	MLE with prior	Regularized estimates	$\hat{\theta}{\mathrm{MAP}}=\arg\max{\theta}\;p(\theta\mid x) =\arg\max_{\theta}\;\big[\log p(x\mid\theta)+\log p(\theta)\big]$
Bayesian Inference	Updates belief with data	Model calibration in biology	$p(\theta\mid x)=\dfrac{p(x\mid\theta)\,p(\theta)}{p(x)},\qquad p(x)=\int p(x\mid\theta)\,p(\theta)\,d\theta$
Hypothesis Test	Decision on effect	Gene expression DE tests	$H_0: \mu_1 = \mu_2$
t-test	Mean difference test	Differential gene expression	$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$
Chi-square test	Goodness of fit	Contingency tables in genomics	$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$
FDR / BH Procedure	Controls multiple testing	GWAS, RNA-seq DE genes	$q_i = \min\left(\frac{p_i m}{i},1\right)$
Bootstrap	Resampling with replacement	CI for small-sample experiments	${\theta}^* = f(X^*)$
Entropy	Uncertainty of distribution	Cell diversity, motif randomness	$H(p) = -\sum_i p_i \log p_i$
Cross-Entropy	Divergence between true & model	Loss for classifiers	$H(p,q) = -\sum_i p_i \log q_i$
KL Divergence	Relative entropy	Compare distributions (healthy vs disease)	$D_{KL}(p\\|q) = \sum_i p_i \log \frac{p_i}{q_i}$
Jensen–Shannon Divergence	Symmetrized KL	Compare embeddings	$D_{JS}(p\\|q) = \tfrac{1}{2}D_{KL}(p\\|m)+\tfrac{1}{2}D_{KL}(q\\|m), m=12(p+q)m=\tfrac{1}{2}(p+q)$
Mutual Information	Shared info between vars	Regulatory network inference	$I(X;Y) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$
Perplexity	Exponential of entropy	Quality of embeddings, models	${Perp}(p) = 2^{H(p)}$
Information Bottleneck	Tradeoff compression vs. relevance	Latent representation of cell states	$\min I(X;Z) - \beta I(Z;Y)$

Optimization and ML Primitives

Component	Definition	In Biology / Computing	Math Variables
Loss Function	Scalar measure of prediction error	Training models for gene expression, structure prediction	$L(\theta) = \frac{1}{N}\sum_i \ell(f_\theta(x_i), y_i)$
Mean Squared Error (MSE)	Average squared error	Regression tasks, kinetics fitting	$L = \frac{1}{n}\sum_i (y_i - \hat{y}_i)^2$
Cross-Entropy Loss	Divergence between distributions	Classification, motif recognition	$L = -\sum_i y_i \log \hat{y}_i$
Hinge Loss	Margin-based loss	Support vector machines, binary classification	$L = \max(0, 1 - y f(x))$
Regularization	Penalty term to avoid overfitting	Control model complexity	$L' = L + \lambda \\|w\\|_p$
L1 Regularization	Promotes sparsity	Feature selection in omics	$\min_{w}\; L(w)+\lambda\\|w\\|{1},\qquad \\|w\\|{1}=\sum_{i=1}^{d}\|w_i\|$
L2 Regularization	Penalizes large weights	Ridge regression, weight decay	$\\|w\\|_2^2 = \sum_i w_i^2$
Gradient Descent	Iterative optimization step	Neural nets, ODE parameter fitting	$\theta_{t+1} = \theta_t - \eta \nabla_\theta L$
Stochastic Gradient Descent (SGD)	Uses minibatches for updates	Large-scale models on bio data	$\theta_{t+1} = \theta_t - \eta \nabla_\theta L(\theta; x_i)$
Momentum	Uses past updates to accelerate	Faster convergence in training	$v_{t+1} = \beta v_t + \nabla_\theta L, \ \theta_{t+1} = \theta_t - \eta v_{t+1}$
Adam Optimizer	Adaptive moment estimation	Standard in DL training	$m_t = \beta_1 m_{t-1} + (1-\beta_1)g_t, \ v_t = \beta_2 v_{t-1}+(1-\beta_2)g_t^2$
L-BFGS	Quasi-Newton optimization	Energy minimization in proteins	Updates use inverse Hessian approximation
Conjugate Gradient	Iterative quadratic solver	Sparse system solvers in genomics	$x_{k+1} = x_k + \alpha_k p_k$
Coordinate Descent	Optimizes one variable at a time	LASSO, constrained models	$x_i^{(t+1)} = \arg\min f(x_1,\dots,x_i,\dots)$
Lagrangian Multipliers	Optimization with constraints	Flux balance analysis	$\mathcal{L}(x,\lambda) = f(x) + \lambda g(x)$
KKT Conditions	Optimality for constrained optimization	Biochemical flux solutions	Stationarity, primal/dual feasibility, complementarity
Proximal Operator	Handles non-smooth penalties	Sparse regression, TV denoising	$\text{prox}_{\lambda f}(v) = \arg\min_x \big(f(x) + \tfrac{1}{2\lambda}\\|x-v\\|^2\big)$
Expectation-Maximization (EM)	Iterative latent variable inference	Mixture models, cell deconvolution	E-step: $Q(\theta)$ , M-step: maximize $Q$
k-Means	Clustering by minimizing within-cluster variance	Cell type clustering	$\arg\min \sum_i \\|x_i - \mu_{c(i)}\\|^2$
Gaussian Mixture Model (GMM)	Soft clustering with Gaussians	Expression distributions	$p(x)=\sum_{k=1}^{K}\pi_k\,\mathcal{N}(x\mid \mu_k,\Sigma_k)$ $\gamma_{ik}=\dfrac{\pi_k\,\mathcal{N}(x_i\mid \mu_k,\Sigma_k)} {\sum_{j=1}^{K}\pi_j\,\mathcal{N}(x_i\mid \mu_j,\Sigma_j)}$
Softmax	Converts scores to probabilities	Classification, attention weights	$\text{softmax}(z_i) = \frac{e^{z_i}}{\sum_j e^{z_j}}$
ReLU	Nonlinear activation	Neural networks	$f(x) = \max(0, x)$
Sigmoid	Squashes to (0,1)	Logistic regression, gating	$\sigma(x) = \frac{1}{1+e^{-x}}$
Tanh	Squashes to (-1,1)	Normalized activations	$\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
Attention Mechanism	Weighted combination of inputs	Protein sequence models, embeddings	$\text{Attn}(Q,K,V) = \text{softmax}\!\left(\frac{QK^T}{\sqrt{d_k}}\right)V$
Contrastive Loss	Brings similar pairs together	Align cell embeddings, protein-ligand	$L = -\log \frac{e^{\text{sim}(x_i,x_j)/\tau}}{\sum_k e^{\text{sim}(x_i,x_k)/\tau}}$
VAE ELBO	Lower bound for latent variable models	Generative scRNA-seq	$\mathcal{L}{\mathrm{ELBO}}(x)= \mathbb{E}{q_{\phi}(z\mid x)}\!\left[\log p_{\theta}(x\mid z)\right] - D_{\mathrm{KL}}\!\big(q_{\phi}(z\mid x)\,\\|\,p(z)\big)$
Diffusion SDE	Forward corruption process	Generative protein design	$dx = f(x,t)\,dt + g(t)\,dW_t$
Graph Laplacian	Encodes connectivity	Protein–protein interaction networks	$L = D - A$
Message Passing	Node embedding updates	GNNs for molecules	$h_v^{(t+1)} = \sigma\!\left(W \cdot \text{AGG}\{h_u^{(t)}:u \in N(v)\}\right)$
Equivariance (SE(3))	Preserves geometric symmetries	Protein structure prediction	$f(Rx) = R f(x)$
Clustering Modularity	Graph-based community detection	Gene co-expression networks	$Q = \frac{1}{2m}\sum_{ij}\Big(A_{ij} - \frac{k_i k_j}{2m}\Big)\delta(c_i,c_j)$

Dynamics (ODEs, PDEs, Stochastic Processes, Control/RL)

Component	Definition	In Biology / Computing	Math Variables
Ordinary Differential Equation (ODE)	Time evolution of variables	Gene circuits, pharmacokinetics	$\frac{dx}{dt} = f(x,t)$
System of ODEs	Multiple interacting variables	Pathways, population dynamics	$\frac{d\mathbf{x}}{dt} = F(\mathbf{x},t)$
Partial Differential Equation (PDE)	Evolution over space + time	Diffusion of molecules, electrophysiology	$\frac{\partial u}{\partial t} = D \nabla^2 u$
Heat Equation	Diffusion PDE	Ion transport, morphogen gradients	$\frac{\partial u}{\partial t} = \alpha \nabla^2 u$
Wave Equation	Propagation dynamics	Nerve impulses, biomechanics	$\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$
Poisson Equation	Potential field equation	Electrostatics in proteins	$-\nabla^2 u = f$
Lotka–Volterra	Predator-prey dynamics	Species competition, host-virus	$\frac{dx}{dt} = \alpha x - \beta xy, \ \frac{dy}{dt} = \delta xy - \gamma y$
Michaelis–Menten Kinetics	Enzyme rate law	Metabolic modeling	$v = \frac{V_{\max}[S]}{K_M + [S]}$
Hill Equation	Cooperative binding	Gene regulation, transcription factors	$\theta = \frac{[L]^n}{K_d^n + [L]^n}$
Mass-Action Kinetics	Reaction rate ∝ concentrations	Systems biology, stoichiometric models	$v = k \prod_i [X_i]^{\nu_i}$
Gillespie Algorithm	Stochastic simulation of reactions	Single-cell variability	Generates trajectories via random exponential waiting times
Markov Chain (transition + distribution update)	Memoryless state transitions	Mutation models, sequence evolution	$\Pr(X_{t+1}=j\mid X_t=i)=P_{ij}, \qquad \pi_{t+1}^{\top}=\pi_{t}^{\top}P$
Hidden Markov Model (HMM)	Latent-state probabilistic model, joint likelihood	Gene finding, protein domains	$p(x_{1:T},z_{1:T})=p(z_1)\, \prod_{t=2}^{T}p(z_t\mid z_{t-1})\, \prod_{t=1}^{T}p(x_t\mid z_t)$
Stochastic Differential Equation (SDE)	Dynamics with noise	Noisy gene expression, molecular motion	$dx = f(x,t)\,dt + g(x,t)\,dW_t$
Ornstein–Uhlenbeck Process	Mean-reverting SDE	Noise in biophysical systems	$dx = \theta(\mu - x)dt + \sigma dW_t$
Random Walk	Successive random steps	Diffusion models, genome scans	$X_{t+1} = X_t + \epsilon_t$
Brownian Motion	Continuous random process	Molecular dynamics	$W(t) \sim \mathcal{N}(0, t)$
Control System	Regulates state of system	Bioreactor control, feedback	$\dot{x} = Ax+Bu, \ y=Cx$
PID Controller	Proportional-integral-derivative control	Lab robotics, process stabilization	$u(t) = K_p e(t) + K_i \int e(\tau) d\tau + K_d \frac{de}{dt}$
Model Predictive Control (MPC)	Optimization-based control	Dynamic bioprocess optimization	$\min_u \sum_{k=0}^N \\|x_k - x^\ast\\|_Q^2 + \\|u_k\\|_R^2$
Markov Decision Process (MDP)	Sequential decision framework	Adaptive experiment design	$\langle S, A, P, R, \gamma \rangle$
Bellman Equation	Recursive value definition	RL-based model design	$V^{}(s)=\max_{a}\Big[r(s,a)+\gamma\sum_{s'}P(s'\mid s,a)\,V^{}(s')\Big]$
Policy Gradient	Optimizes expected reward	Reinforcement learning in biology	$\nabla_{\theta}J(\theta)= \mathbb{E}{\pi{\theta}}\!\left[\sum_{t} \nabla_{\theta}\log \pi_{\theta}(a_t\mid s_t)\,A_t\right]$
Advantage Estimation (GAE)	Variance-reduced estimator	Efficient policy training	$A_t = \sum_{l=0}^\infty (\gamma \lambda)^l \delta_{t+l}$
PPO (Proximal Policy Optimization)	RL with clipped objective	Safe training in bio RL models	$L^{CLIP} = \mathbb{E}[\min(r_t(\theta)A_t, \text{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t)]$
SAC (Soft Actor-Critic)	RL with entropy maximization	Exploration in protein design	$J(\pi)=\sum_{t}\mathbb{E}_{(s_t,a_t)\sim \pi}\!\left[ r(s_t,a_t)+\alpha\,\mathcal{H}\big(\pi(\cdot\mid s_t)\big)\right],\quad \mathcal{H}(\pi(\cdot\mid s))=-\!\int \pi(a\mid s)\log \pi(a\mid s)\,da$

Signal Processing, Numerical Methods & HPC

Component	Definition	In Biology / Computing	Math Variables
Convolution	Weighted overlap of functions	Motif scanning, microscopy filtering	$(f*g)(t) = \int f(\tau)g(t-\tau)\,d\tau$
Correlation	Similarity via shifted overlap	Template matching in sequences	$(f \star g)(t) = \int f(\tau)g(t+\tau)\,d\tau$
Fourier Transform	Decomposes into frequencies	MRI k-space, EEG	$\hat f(\omega) = \int f(t) e^{-i\omega t} dt$
Discrete Fourier Transform (DFT)	Finite-sample version	Sequence periodicity detection	$X_k = \sum_{n=0}^{N-1} x_n e^{-i2\pi kn/N}$
Fast Fourier Transform (FFT)	Efficient DFT algorithm	Bio-signal analysis at scale	$O(N \log N)$ complexity
Wavelet Transform	Time–frequency decomposition	Microscopy image denoising	$W(a,b) = \int f(t)\psi\!\left(\frac{t-b}{a}\right)dt$
Radon Transform	Line integrals of function	CT reconstruction	$Rf(\theta,s) = \int f(x,y)\,\delta(s-x\cos\theta-y\sin\theta)\,dxdy$
Filter (Low/High-pass)	Signal smoothing or sharpening	Noise reduction in time series	Frequency cutoff: $H(\omega)$
Wiener Filter	Linear MMSE estimator	Signal denoising	$H(\omega) = \frac{S_{xx}(\omega)}{S_{xx}(\omega)+S_{nn}(\omega)}$
Kalman Filter	Recursive state estimator	Tracking cell motion	$\textbf{Predict:}\quad \hat x_{k\mid k-1}=A\,\hat x_{k-1\mid k-1}+B\,u_k,\qquad P_{k\mid k-1}=A\,P_{k-1\mid k-1}A^{\top}+Q \textbf{Update:}\quad K_k=P_{k\mid k-1}H^{\top}\big(H P_{k\mid k-1}H^{\top}+R\big)^{-1} \\ \hat x_{k\mid k}=\hat x_{k\mid k-1}+K_k\big(y_k-H\hat x_{k\mid k-1}\big),\qquad P_{k\mid k}=(I-K_k H)P_{k\mid k-1}$
Particle Filter	Sequential Monte Carlo	Nonlinear/noisy tracking	Approximate posterior via particles
Total Variation (TV) Denoising	Penalizes gradient magnitude	Microscopy deblurring	$\min_x \\|x-y\\|^2 + \lambda \\|\nabla x\\|_1$
Compressed Sensing	Recovery from undersampling	Accelerated MRI	$\min_x \\|x\\|_1 \ \text{s.t.}\ Ax=b$
Finite Difference	Approx derivative by discretization	PDE solvers	$\frac{\partial u}{\partial x} \approx \frac{u(x+h)-u(x)}{h}$
Finite Element Method (FEM)	Domain discretization into elements	Biomechanics, electrophysiology	Weak form: $\int_\Omega \nabla u \cdot \nabla v \,dx$
Finite Volume Method	Conserves fluxes per cell	Transport models in tissues	$\frac{d}{dt}\int_\Omega u\,dx + \int_{\partial\Omega} F\cdot n \,ds = 0$
Monte Carlo Integration	Random sampling for integrals	Partition functions, uncertainty	$I \approx \frac{1}{N}\sum f(x_i)$
Importance Sampling	Weighted MC estimates	Rare-event modeling	$I = \frac{1}{N}\sum \frac{f(x_i)}{q(x_i)}$
Quasi-Monte Carlo	Low-discrepancy sequences	Faster convergence for high-dim integrals	Uses Sobol / Halton sequences
Automatic Differentiation	Programmatic derivative	Training ML models	Forward & reverse mode $\frac{dy}{dx}$
Backpropagation	Reverse-mode autodiff	Neural nets in bio	$\frac{\partial L}{\partial w}$ via chain rule
Roofline Model	Performance vs. arithmetic intensity	Kernel optimization	FLOPs/byte tradeoff
Amdahl’s Law	Parallelism speedup bound	Multi-core scaling limits	$S = \frac{1}{(1-p)+p/N}$
Gustafson’s Law	Scaling efficiency with workload	HPC bio pipelines	$S = N - (N-1)(1-p)$
Memory Bandwidth Limit	Bytes/sec bottleneck	GPU genomics kernels	Throughput = min(compute, memory BW)
SIMD / GPU Parallelism	Single-instruction, many data	K-mer counting, alignment	Vector ops per cycle
Sparse Matrix Ops	Efficient storage/computation	Genome graphs, scRNA matrices	Formats: CSR, COO

Bioinformatics and Sequence Mathematics

Component	Definition	In Biology / Computing	Math Variables
k-mer	Substring of length kk	Genome comparison, sequence hashing	$s[i:i+k]$
Jaccard Index	Set similarity measure	Genome sketching, assembly comparison	$J(A,B)=\dfrac{\|A\cap B\|}{\|A\cup B\|}$
MinHash	Fast Jaccard approximation	Large-scale sequence similarity	Randomized hashing of k-mers
Count-Min Sketch	Probabilistic frequency table	Streaming k-mer counts	$\tilde f(i) = \min_j C[j,h_j(i)]$
Hamming Distance	Number of differing positions	DNA barcode error correction	$d_H(x,y) = \sum_i [x_i \ne y_i]$
Levenshtein Distance	Edit distance (insert/del/sub)	Sequence alignment	Minimum edits to transform x→yx \to y
Smith–Waterman	Local alignment DP	Short sequence homology	$F_{i,j} = \max\{0, F_{i-1,j-1}+s, F_{i-1,j}-d, F_{i,j-1}-d\}$
Needleman–Wunsch	Global alignment DP	Genome alignment	$F_{i,j} = \max\{F_{i-1,j-1}+s, F_{i-1,j}-d, F_{i,j-1}-d\}$
Gotoh Algorithm	Alignment with affine gaps	Realistic indel scoring	Gap cost $g + k e$
Substitution Matrix	Scoring amino acid swaps	BLOSUM, PAM	$S(a,b) = \log \frac{P(a,b)}{P(a)P(b)}$
Position Weight Matrix (PWM)	Motif probability matrix	TF binding site prediction	P $_{i}(b)$ , where $b$ is base
Hidden Markov Model (Profile HMM)	Motif/sequence family model	Protein domains	Transition + emission probabilities
Suffix Array	Sorted suffix positions	Fast substring search	$SA$ = sorted indices of suffixes
Suffix Tree	Tree of suffixes	Genome indexing	Nodes = substrings
Burrows–Wheeler Transform (BWT)	Reversible string transform	Basis of read mappers	Last column of sorted rotations
FM-Index	Compressed substring index	Read mapping	Supports $O(m)$ pattern matching
De Bruijn Graph	k-mer graph structure	Genome assembly	Nodes = k-1-mers, edges = k-mers
Eulerian Path	Traverses each edge once	Assembly from k-mers	Exists if in-degree = out-degree
Hamiltonian Path	Traverses each node once	Overlap-layout assembly	NP-hard
Phred Score	Log-scaled error probability	Sequencing quality	$Q = -10\log_{10} p$
Codon Usage Bias	Frequency of synonymous codons	Expression optimization	CAI, tAI formulas
Codon Adaptation Index (CAI)	Expression potential metric	Gene design	$\text{CAI} = \left(\prod_i w_i\right)^{1/L}$
tRNA Adaptation Index (tAI)	Translation efficiency score	Synthetic biology	Based on tRNA availability
GC Content	Fraction of G+C bases	Genomic stability	$\frac{G+C}{A+T+G+C}$
k-mer Spectrum	Histogram of k-mer counts	Detect heterozygosity, repeats	$f(k)$ distribution
Sequence Entropy	Information content of sequence	Motif conservation	$H = -\sum p_b \log p_b$
Motif Scanning (Convolution)	PWM convolution across genome	Regulatory site finding	$Score = \sum_i \log P_i(x_i)$
BLAST Scoring	Heuristic local alignment	Homology search	Uses seed-and-extend + substitution matrices
Phylogenetic Tree	Evolutionary tree model	Ancestral inference	Distance or likelihood based
Felsenstein Pruning Algorithm	Likelihood computation on tree	Phylogenetic likelihoods	Dynamic programming over nodes

Systems Biology, Structural Biology & Population Genetics

Component	Definition	In Biology / Computing	Math Variables
Stoichiometric Matrix (S)	Encodes reaction network	Flux balance analysis (FBA)	$S \cdot v = 0$
Flux Balance Analysis (FBA)	Linear optimization on S	Metabolic pathway prediction	$max c^T v \ \text{s.t.}\ S v = 0, \ l \le v \le u$
Flux Variability Analysis (FVA)	Range of feasible fluxes	Robustness of metabolism	Optimizes min/max viv_i under FBA constraints
Parsimonious FBA (pFBA)	Minimizes total flux	Efficient metabolic solutions	$\min_{v}\;\sum_{i}\|v_i\| \quad \text{s.t.}\quad S v=0,\; v_{\min}\le v\le v_{\max}$
Metabolic Control Analysis	Quantifies control coefficients	Sensitivity in pathways	$C^J_i = \frac{\partial \ln J}{\partial \ln E_i}$
Michaelis–Menten	Enzyme kinetics	Reaction velocity	$v = \frac{V_{\max}[S]}{K_M+[S]}$
Hill Equation	Cooperative binding	TF–DNA regulation	$\theta = \frac{[L]^n}{K_d^n+[L]^n}$
Mass-Action Law	Rate ∝ reactant concentrations	Reaction network modeling	$v = k\prod_i [X_i]^{\nu_i}$
Arrhenius Equation	Temp dependence of rate	Biochemical kinetics	$k = A e^{-E_a/RT}$
Eyring Equation	Transition state theory	Reaction thermodynamics	$k = \frac{k_B T}{h} e^{-\Delta G^\ddagger /RT}$
Gibbs Free Energy	ΔG predicts spontaneity	Protein folding, binding	$\Delta G = \Delta H - T\Delta S$
Binding Equilibrium	Ligand–receptor affinity	Protein–drug interactions	$K_d = \frac{[P][L]}{[PL]}$
ΔG–Kd Relation	Thermodynamic link	Quantifying binding strength	$\Delta G = -RT \ln K_d$
Force Fields	Energy functions in MD	Protein simulations	$E = E_{bond}+E_{angle}+E_{torsion}+E_{vdW}+E_{elec}$
Lennard–Jones Potential	van der Waals model	Molecular packing	$V(r)=4\epsilon[(\sigma/r)^{12}-(\sigma/r)^6]$
Coulomb’s Law	Electrostatic interactions	Charged biomolecules	$F = \frac{kq_1 q_2}{r^2}$
Ewald/PME Summation	Long-range electrostatics	Protein MD	Splits short/long-range terms
Root Mean Square Deviation (RMSD)	Structure difference metric	Protein structure evaluation	$\text{RMSD} = \sqrt{\frac{1}{N}\sum_i \\|x_i-y_i\\|^2}$
TM-score	Protein structural similarity	Structure prediction accuracy	$\text{TM} = \max\left(\frac{1}{L}\sum \frac{1}{1+(d_i/d_0(L))^2}\right)$
Contact Map	Binary residue contacts	Folding, docking models	$C_{ij} = [d_{ij} < \delta]$
Ramachandran Plot	φ–ψ torsional space	Protein conformational analysis	Allowed regions of $\phi, \psi$
Rotamer Library	Discrete side-chain conformations	Protein modeling	Probabilities over torsion states
Free Energy Perturbation (FEP)	ΔΔG between states	Binding affinity prediction	$\Delta G = -k_BT \ln \langle e^{-\Delta U/k_BT}\rangle$
Thermodynamic Integration (TI)	Computes ΔG via λ interpolation	Drug design	$\Delta G = \int_0^1 \langle \partial U/\partial \lambda \rangle_\lambda d\lambda$
MBAR	Multi-state free-energy estimator	Protein/ligand ΔΔG	Weighted combination of samples
Hardy–Weinberg Equilibrium	Allele frequency model	Population genetics	$p^2+2pq+q^2=1$
Wright–Fisher Model	Genetic drift in finite pops	Allele frequency variance	Binomial sampling of alleles each gen
Moran Model	Overlapping-gen population model	Drift, fixation	One birth + one death per step
Coalescent Theory	Backward-time genealogy	Ancestral allele inference	Distribution of coalescent times
Fixation Probability	Probability allele becomes fixed	Selection vs drift	$P_{\text{fix}} \approx \frac{1-e^{-2s}}{1-e^{-2Ns}}$
Substitution Models	Models nucleotide changes	Phylogenetics	JC69, HKY, GTR matrices
Felsenstein Pruning	Likelihood on trees	Phylogenetic inference	Recursive likelihood computation

Evaluation, Scaling & Cryptography

(includes some quantum)

Component	Definition	In Biology / Computing	Math Variables
Accuracy	Fraction of correct predictions	Classifier evaluation	$\frac{TP+TN}{TP+FP+FN+TN}$
Precision	Correct positives / all positives	Gene variant calling	$\frac{TP}{TP+FP}$
Recall (Sensitivity)	True positives / actual positives	Rare mutation detection	$\frac{TP}{TP+FN}$
Specificity	True negatives / actual negatives	Diagnostic screening	$\frac{TN}{TN+FP}$
F1 Score	Harmonic mean of precision & recall	Balancing bio classifier performance	$2\cdot\frac{PR}{P+R}$
ROC Curve / AUC	Tradeoff sensitivity vs specificity	Diagnostic classifiers	$AUC$ = area under curve
PR Curve / AUC	Precision–recall tradeoff	Imbalanced omics data	Area under PR curve
Matthews Correlation (MCC)	Balanced measure even with imbalance	DNA classification	$\frac{TP\cdot TN - FP\cdot FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}$
Brier Score	Calibrated probability error	Probabilistic predictions	$BS = \frac{1}{N}\sum (p_i - y_i)^2$
Calibration (Temp Scaling)	Adjusts softmax confidence	Protein function prediction	$q_i = \text{softmax}(z_i/T)$
Conformal Prediction	Distribution-free prediction intervals	Genomic risk scores	$[L(x), U(x)] with coverage 1−α1-\alpha$
Aleatoric Uncertainty	Intrinsic randomness	Sequencing noise	Modeled in likelihood variance
Epistemic Uncertainty	Model ignorance	Limited training data	Ensembles, Bayesian NNs
Learning Curve	Error vs dataset size	Scaling genomic models	$L(N) \approx a N^{-\alpha} + b$
Power Law (Scaling Law)	Performance vs compute/data	Deep learning in bio	$L(C) = k C^{-\beta} + \epsilon$
Chinchilla Law	Optimal compute–data balance	Training large bio models	Loss scales with tokens ∝ $C^{1/2}$
Wright’s Law	Cost falls with production	Sequencing costs	$C(n) = C_0 n^{-\alpha}$
Queueing Model (Little’s Law)	Throughput relation	Bio pipeline scheduling	$L = \lambda W$
Sensitivity Analysis	Effect of parameter variation	Bioprocess robustness	$S_i = \frac{\partial y}{\partial \theta_i}$
Cryptographic Hash	One-way function	Genomic data integrity	$h(x)$
Homomorphic Encryption (HE)	Compute on ciphertexts	Privacy-preserving genomics	$E(a+b) = E(a)\cdot E(b)$
Lattice-based Crypto (RLWE)	Hard lattice problem	Secure bio models	$a s + e \pmod{q}$
CKKS Scheme	Approximate HE for reals	Encrypted ML inference	Supports $+$ , $\times$ on ciphertexts
Noise Budget	Error growth in HE ops	Bio AI on encrypted data	Ciphertext validity bound
Quantum Operator Algebra	Linear operators on Hilbert space	Quantum chemistry models	( \hat H
Spectral Decomposition	Expanding in eigenbasis	Quantum Hamiltonians, protein folding	$A = \sum_i \lambda_i v_i v_i^T$
Tensor Product	Composite quantum states	Multi-particle biology	$\|\psi\otimes\phi\rangle =\|\psi\rangle\otimes\|\phi\rangle =\begin{bmatrix}\psi_1\\\psi_2\end{bmatrix}\otimes \begin{bmatrix}\phi_1\\\phi_2\end{bmatrix} =\begin{bmatrix} \psi_1\phi_1\\ \psi_1\phi_2\\ \psi_2\phi_1\\ \psi_2\phi_2 \end{bmatrix}$
Density Matrix	Mixed state representation	Open system biology	$\rho=\sum_{i}p_i\,\|\psi_i\rangle\langle\psi_i\|$
von Neumann Entropy	Entropy of quantum state	Quantum biology analogs	$S(\rho) = -\text{tr}(\rho \log \rho)$

Mathematical Tools

Calculus + Linear Algebra

Probability, Statistics, Information Theory

Optimization and ML Primitives

Dynamics (ODEs, PDEs, Stochastic Processes, Control/RL)

Signal Processing, Numerical Methods & HPC

Bioinformatics and Sequence Mathematics

Systems Biology, Structural Biology & Population Genetics

Evaluation, Scaling & Cryptography