quant.bio PREVIEW A mathematical way to think about biology main site | handouts | @ + ■ ■ ■ ■ ■ - | + ■ ■ ■ ■ ■ -

Physical sciences perspectives provide biological insights

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 “ Excellent site for both basic and advanced lessons on applying mathematics to biology.

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I. Aspects of biology can be quantitatively modeled

 💡 The appendix includes a discussion of algebra, precalculus, and calculus.

Rates of change can be derived from propositions and used to predict accrued change in biology

Some quantities in biological systems are hypothesized to change over time in apparently stochastic ways

Stochasticity

1 Incommensurate periods
2 Practical unpredictability
3 Fundamental indeterminism
4 Illustrating memory-free (Markov) processes

Some stochastic processes can be investigated using deterministic reaction kinetics models,

Cartoon model of protein dynamics

5 Translation and degradation occur over time
6 Differential equation and flowchart
7 Qualitative graphical solution to differential equation
8 Analytic solution and rise time

Law of mass action

9 Oversimplified derivation of law of mass action
10 Oversimplified cooperativity and Hill functions
11 Bistability
$$\frac{\mathrm{d}R}{\mathrm{d}t} = k x_1 x_2 \cdots x_N$$
[1] w3 Iwasa et al. Molecular Flipbook open-source biomolecular animation software

not just at molecular scales, but also at other scales, e.g., at population scales

Evolutionary game theory (EGT) I

12 Population dynamics with interactions
13 Tabular game theory (comparative statics)

EGT II: Relating dynamics and statics

14 Evolution resulting from repeated game play
15 Relationship between time and sophisticated computation

EGT III: Training and validating population dynamics equations

[2] OA Liao and Tlsty, Interface Focus 4:20140037 (2014)
[3] OA Liao and Tlsty, Interface Focus 4:20140038 (2014)

Data can be summarized and compared with theoretical probability distributions

Varied data and variability can be summarized

Statistics

16 Probability distributions and averages
17 Identities involving averages
18 Dispersion and variance
19 Statistical independence
20 Identities following from statistical independence

When a model is proposed, the probability of drawing a value of a statistic can be derived

Probability

21 Bernoulli trial
22 Binomial distribution
23 Poisson limit

Central limit theorem

24 Stirling's approximation
25 CLT a: Statement of central limit theorem
26 CLT b: Optional derivation (special case)
27 CLT c: Properties of Gaussian distributions

Universality of normal distributions

28 Physics labs
29 Log-normal biology

[4] w3 Normal distribution§Occurrence, Wikipedia (accessed 2014 October 2).

Subjective criteria can be used to proclaim a value of a statistic (given a model) low enough to "signify" rejection of a hypothesis

Uncertainty propagation

31 Sample estimates
32 Square-root of sample size ($\sqrt{n}$) factor
33 Error bar "overlap" criterion
34 Illusory sample size

Sample variance curve fitting

35 Chi-squared ($\chi^2$)
36 Minimizing $\chi^2$
37 Checklist for undergraduate curve fitting

$$\chi^2 \sim M - N_{PARAM}$$
PDF Sample-variance curve fitting exercise for MatLab
[5] w3 $\chi^2$ table from M. Akritas's course at Penn State

Choosing criteria based on commercial objectives

Even though academic tradition is a major contributor to the adoption of criteria for declaring statistical "significance," for example, the arbitrary p < 0.05 criterion, it is possible to choose criteria in a more rational manner. For example, a pharmaceutical company could compare potential research and legal costs with potential profits to conclude that it can only afford to move on to Phase III trials for experimental therapies that rule out a null hypothesis at a p < 0.01 level during Phase II trials.

Guard against using quantitative relationships to misinterpret noise

Statistical analysis can be misunderstood, leading to misinterpretation of data

Misinterpreted as certifications of reproducibility

95% confidence intervals do not necessarily thread the means of their associated parent distributions 95% of the time. Confidence intervals can be meaningless when derived from faulty assumptions.

Misinterpreted as certifications of hypothesis "truth"

Rejecting models having p < 0.1 does not ensure that 90% of surviving models are valid. Human imagination easily contrives a large number mutually incompatible models, each of which is consistent with a given set of experimental data at the p > 0.1 level. This consistency cannot logically guarantee that 90% of the models are valid since, by construction, most of them are false.

[6] PDF Nuzzo, Nature 506: 150-152 (2014).
[7] PDF Goodman, Epidemiology 12: 295-297 (2001).

Probabilistic dynamics can be predicted from probabilistic hypotheses

Distributions and trajectories can be calculated for probabilistic dynamic systems

Stochastic dynamics

38 Master equation
39 Stochastic simulation algorithm

Poissonian copy numbers

40 Stochastic synthesis and deterministic degradation

Accrued change in linear/ish systems can be studied by finding eigenvectors

Linear dynamics in discrete time steps can be analyzed using eigenvectors

Linear algebra I

43 Vectors
44 Operators
45 Intuitive guess for solution of teaser
46 Solution based on eigenvalue-eigenvector analysis
$$\det(\mathbb{A} - \lambda \mathbb{I}) = 0$$

Biological example

47 Cartoon quasispecies model

Vector rotation

48 Euler's number II: Euler's formula
49 Linear algebra II: Rotation matrix
50 Complex eigenvalues

Linear dynamics in continuous time can be analyzed using eigenvectors

Differential equations (DEs)

51 Rough introduction to direction fields and of numerical integration

mRNA transcription-protein translation model

52 Transcription-translation model
53 Eigenvector-eigenvalue analysis
54 Cribsheet of linear stability analysis

Adaptation and the incoherent feed-forward loop motif

56 Cribsheet of almost linear stability analysis

Eigenvector analysis is just one example of a use of quiver fields

Quiver fields can be used to describe oscillations in terms of DEs, time delays, and excitability

Oscillations

57 Romeo and Juliet
58 Twisting nullclines
59 Time delays
60 Stochastic excitation

The ability of a system of DEs to produce solutions that resemble data does not rule out alternative descriptions of dynamics.

[8] PDF Ferrell, Jr., Tsai, and Yang, Cell, 144: 874-885 (2011).

II. Quantitative biology provides insights for human society

Quantitative reasoning can be applied to oncology

Physical and biological scientists are working together to develop insights into cancer biology and treatment

Introduction to physical oncology

61 Introduction to the PSOC Network

Dynamic heterogeneity can be used to study therapeutic dose-scheduling

Dynamic heterogeneity and the metronomogram

62 Stochastic biochemistry
63 Phenotypic interconversion
64 Metronomogram
$$f_S(\Delta t) > f_P(\Delta t)$$
[9] OA Liao, Estévez-Salmerón, and Tlsty Phys. Biol. 9:065005 (2012)
[10] OA Liao, Estévez-Salmerón, and Tlsty Phys. Biol. 9:065006 (2012)

Statistics of rare events can provide insights into the establishment of metastases

[11] OA Cisneros and Newman Phys. Biol. 11:046003 (2014)

(In progress)

III. The structure of biological systems and the behaviors of systems and their parts influence each other

System behaviors depend on topological organization of component interactions

Local interactions can generate population dynamics qualitatively distinct from those produced by well-mixed models

Modeling spatially-resolved systems

65 Deterministic cellular automata

[12] w3 A. Aktipis's agent-based modeling resources
[13] w3 A. Kaznatcheev's introduction to the Ohtsuki-Nowak transform

The success of a non-spatial model in mimicking aggregate population dynamics does not rule out a spatially-resolved model.

Understanding consequences of organizational structure provides insight into biological systems

Spatial organization of biological heterogeneity affects dynamics

[14] PDF Kerr et al. Nature 418:171--174 (2002)
[15] PMC Macklin et al. J. Theor. Biol. 301: 122–140 (2012).

(In progress)

IV. Microstates of a system can be explored when microstates of the universe are explored

Eventually, all accessible microstates are equally accessed

Statistics can be calculated for small subparts of much larger universes at thermal equilibrium

If closed universes (eventually) explore all accessible states equally, then small subsystems can be studied using partition functions

Equilibrium statistical mechanics

66 Accessible states are eventually equally visited
67 Notating configurations of a system
68 Distribution of energy between small system and big bath
69 Calculating average properties of small subsystems

$$Z = \sum_{i}^{}{g(E_i) e^{-\frac{E_i}{\tau}}}$$

At finite temperature, even an ideal polymer chain exhibits "entropic elasticity"

Partition function and expected elongation of ideal chain

70 Introduction to model
71 Hamiltonian and partition function
72 Expectation of energy and elongation

Statistical mechanics is used to describe the conformations of DNA

[16] PDF Bustamante et al. Curr. Opin. Struct. Biol. 10(3):279--285 (2000)

Microscopic reversibility implies macroscopic irreversibility

Macroscopic irreversibility depends on the ratio of kinetically accessible volumes of phase space associated with two macrostates

DISCLAIMER: The following video chapter is a simplified introduction that can be used in algebra-based high school physics classes. This is not a replacement for the references below.

Fluctuation theorem and effective ratio of phase space volumes

73 Microstates of the universe are explored over time
74 Microscopic reversibility
75 Ratio of phase space volumes
76 Kinetically accessible volumes of phase space

$$\frac{P_{C_2 \rightarrow C_1}}{P_{C_1 \rightarrow C_2}} = \frac{\mathrm{size}(C_1)}{\mathrm{size}(C_2)} = e^{-\frac{\Delta Q}{\tau}}$$
[17] PDF Crooks Excursions in Statistical Physics. PhD thesis, University of California, Berkeley (1999)
[18] OA England J. Chem. Phys. 139:121923 (2013)
[19] w3 England Perimeter Institute Recorded Seminar Archive 139:14090073 (2014)

V. Appendix

Quantitative relationships can be reasoned from propositions

A quantity represents a specified outcome from an ordered sequence of possibilities

Numbers

78 Bose-Einstein condensate
79 Visual representations
80 Infinity is not a number

Quantities can be related

Algebra

81 Variables
82 Functions
83 Composite functions
84 Inverse functions
85 Square-root function and $i$

Quantitative relationships can be derived

Geometry

87 Euclidean
88 Sine and cosine
89 Approximating $\pi$
90 Right triangles and trigonometric identities

Summation

91 Sigma notation
92 Introduction to infinite series

Combinatorics

93 Permutations and factorials
94 Combinations
95 Binomial theorem

Change accrued is related to rates and durations of change

Limits

96 Limit of a function
97 Improper (infinite) limits
98 Limits "at" infinity
99 Infinite limits "at" infinity
100 Limits do not always exist
101 $\epsilon$-$\delta$ proof of a limit of a linear function

Quantities can change at quantifiable rates

Differentiation

102 Derivatives and differentials
103 Power rule
104 Chain rule
105 Products and quotients
106 Sinusoidal functions
107 Partial differentiation

Power series representations

108 Second derivative describes curvature
109 Determining power series terms
110 Power series for sine
111 Decimal approximation for $\pi$

Change can be accrued

Integration

112 Area under a curve
113 Rate of change of accrued area is the height
114 Integral over interval equals function evaluated at bounds
115 Change of variables rule
116 Separation of variables
$$\int_{a}^{b} \frac{\mathrm{d}f}{\mathrm{d}x} \mathrm{d}x = f(b) - f(a)$$
Calligraphy manifests the fundamental theorem of calculus:
1. Thicker strokes cover paper at faster rates.
2. Painted area depends both on length through which and on thickness(es) at which brush was drawn.

Euler's number I

117 Compound interest
118 e to the zero
119 Exponent multiplication identity