PRINCIPLES OF COMPUTER SCIENCE MATHEMATICS AND PHYSICS APPLIED TO BIOTECHNOLOGY 1

Module PRINCIPLES OF COMPUTER SCIENCE AND MATHEMATICS APPLIED TO BIOTECHNOLOGY

Expected Learning Outcomes in line with the UN 2030 Agenda for Sustainable Development

By the end of the course, students will be able to:

1. Mathematical and digital competencies

   • Apply fundamental concepts of mathematics (functions, derivatives, integrals, probability, and statistics) to describe and analyze biological phenomena.

   • Use digital tools and spreadsheets for the management and processing of experimental data.
     (SDG 4 – Target 4.4: increase technical and scientific skills for employment and innovation)

2. Data analysis and scientific method

   • Process real biological datasets through descriptive and inferential statistical methods.

   • Critically interpret numerical and statistical results to draw reliable conclusions.
     (SDG 9 – Target 9.5: enhance scientific research and technological capacity)

3. Modeling of biological systems

   • Apply mathematical models (differential equations, regressions, growth models) to biological and biotechnological processes.

   • Assess scenarios and predictions relevant to biotechnological research and the management of biological resources.
     (SDG 3 – Target 3.b: support research and development of biotechnological solutions for health; SDG 15 – Target 15.5: reduce biodiversity loss)

4. Transversal skills and sustainability

   • Use quantitative tools to contribute to innovative solutions in biotechnology with an emphasis on environmental and economic sustainability.

   • Promote a critical approach to the use of data and digital technologies, fostering responsible and inclusive practices.
     (SDG 12 – Target 12.2: sustainable use of natural resources; SDG 4 – Target 4.7: ensure knowledge and skills needed to promote sustainable development)

In-person lectures
Lectures will be conducted in person, in accordance with current regulations.

Information for students with disabilities and/or learning disorders (DSA)
To ensure equal opportunities and compliance with current legislation, students with disabilities or DSA may request an individual meeting to plan any compensatory and/or dispensatory measures, in line with the learning objectives and their specific needs.
It is also possible to contact the CInAP faculty representative for further information and support.

Students are expected to have:

1. Basic mathematical skills

• Elementary arithmetic: operations with integers, fractions, and decimals.

• Elementary algebra: first- and second-degree equations and inequalities, simple linear systems.

• Basic plane and solid geometry: lines, planes, geometric figures and their essential properties.

• Elementary functions: concept of function, graphs of linear, quadratic, power, exponential, and logarithmic functions.

• Basic trigonometry (sine, cosine, tangent).

2. General logical and scientific skills

• Ability for logical and abstract reasoning.

• Basic knowledge of natural sciences (high-school level biology and chemistry) to understand examples and applications.

3. Basic computer skills

• Essential use of a computer: file and folder management, word processing, internet navigation.

• Basic familiarity with spreadsheets (data entry, simple arithmetic operations).

1 – Mathematical Foundations

• Sets, elementary logic, and basic operations.

• Real numbers, absolute value, and the triangle inequality.

• Powers, roots, and logarithms.

• Percentages, proportions, and growth rates (applications to biological data).

• Summations, binomial coefficients, arithmetic and weighted means.

• Elementary equations, inequalities, and systems.

2 – Functions and Graphs

• Cartesian plane and graphical representation of experimental data.

• Concept of function and main families of functions (power, exponential, logarithmic, trigonometric).

• Sequences and limits.

• Continuity and asymptotic behavior of functions.

• Applications to biological growth and decay models.

3 – Differential and Integral Calculus

• Concept of derivative: geometric and physical interpretations.

• Derivatives of elementary functions and calculation rules.

• Maxima and minima: applications to optimization problems in biology (e.g., population growth, optimal dosages).

• Definite and indefinite integrals: basic concepts and applications to cumulative biological quantities (e.g., concentration over time).

• Introduction to differential equations with applications to dynamic models in biotechnology.

4 – Linear Algebra and Models

• Vectors and matrices, fundamental operations.

• Linear systems and methods of solution.

• Eigenvalues, eigenvectors, and population dynamics (introduction to the Perron-Frobenius theorem).

5 – Probability and Statistics

• Concept of probability, events, and equiprobable models.

• Conditional probability, independence, and Bayes’ theorem.

• Discrete and continuous random variables; main distributions (binomial, Poisson, normal, uniform, exponential).

• Law of large numbers and central limit theorem.

• Descriptive statistics: frequency tables, histograms, measures of central tendency and variability, boxplots.

• Bivariate statistics: correlation and linear regression.

• Inferential statistics: point and interval estimation, hypothesis testing (Z-test, t-test, χ²-test, proportion tests).

6 – Computer Science and Digital Tools

• Introduction to spreadsheets (Excel, LibreOffice Calc or equivalents).

• Management of biological datasets: importing, cleaning, and formatting data.

• Automatic calculations, formulas, and functions.

• Statistical analysis with spreadsheets: mean, variance, regression, hypothesis testing.

• Creating scientific charts for data visualization and communication.

• Introduction to basic programming functions in spreadsheets (e.g., IF, LOOKUP, simple macros).

7 – Interdisciplinary Applications

• Mathematical modeling in biotechnology: microbial growth, enzyme kinetics, molecular diffusion.

• Data management: from laboratory results to statistical representation.

• Case studies: analysis of real experimental datasets from biological research.

The exam consists of a single written test, and the evaluation will be assigned according to the following criteria:

Grade 29–30 with honors

• In-depth knowledge of the subject

• Ability to integrate and critically analyze the situations presented

• Independent resolution of complex problems

• Excellent communication skills and mastery of language

Grade 26–28

• Good knowledge of the subject

• Ability to perform clear and critical analysis of situations

• Fairly independent resolution of complex problems

• Clear presentation and appropriate language

Grade 22–25

• Fair knowledge, limited to the main topics

• Critical analysis not always consistent

• Fairly clear presentation with acceptable use of language

Grade 18–21

• Minimal knowledge of the subject

• Limited ability to integrate and critically analyze situations

• Sufficiently clear presentation, with weak language proficiency

Exam not passed

• Lack of knowledge of the main contents

• Very limited or no ability to use specific terminology

• Inability to independently apply acquired knowledge

Compensatory and dispensatory measures

To ensure equal opportunities and in compliance with current regulations:

• Interested students may request a personal meeting to arrange possible compensatory and/or dispensatory measures, based on the learning objectives and their specific needs.

• Students may also contact the CInAP representative (Center for Active and Participatory Integration – Services for Disabilities and/or Specific Learning Disorders) of their Department.

Examples of Basic Mathematics Questions

1. Solve the following equation:

   $$2x^2-5x+3=0$$

2. Consider the function $f(x) = e^{-0.2x}$.

   • Sketch the graph qualitatively.

   • Calculate the first derivative and discuss its sign.

   • Interpret the behavior of the function as a model of biological decay (e.g., the concentration of a drug in the blood).

Examples of Differential and Integral Calculus Questions

3. A bacterial population grows according to the logistic law:

   $$P(t)=\frac{1000}{1+9e^{-0.5t}}$$

   • Calculate $P(0)$ and $\lim_{t \to \infty} P(t)$.

   • Determine the initial growth rate (derivative at $t=0$).

4. Calculate the integral:

   $${\int }_0^1(3x^2+2x)dx$$

   and interpret the result as the “cumulative quantity” in a biological process.

Examples of Linear Algebra Questions

5. Solve the following linear system:

   $$\{\begin{array}{l}x+y+z=3\\ 2x-y+z=0\\ x+2y-z=1\end{array}$$

6. A matrix is given by

   $$A=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]$$

   • Calculate eigenvalues and eigenvectors.

   • Briefly explain how these concepts apply to the study of dynamic models in biotechnology.

Examples of Probability and Statistics Questions

7. In a laboratory, 200 cells are analyzed. It is observed that 40 have a genetic mutation.

   • Calculate the sample proportion and a 95% confidence interval for the proportion.

8. Suppose that the degradation time of a substance follows an exponential distribution with parameter $\lambda = 0.2$.

   • Calculate the probability that the substance degrades in less than 5 hours.

9. A dataset contains the following concentration values (mg/L):
   4.1, 3.8, 4.5, 4.2, 3.9, 4.0

   • Enter the data into a spreadsheet.

   • Calculate the mean, variance, and standard deviation.

   • Represent the data using a histogram.