Selection exam TU Delft

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Questions and Answers

Explain how the influx and outflux parameters are used in the basic differential equations for modeling systems like a growing bacterial strain.

Influx represents factors that increase the system parameter, such as population growth or contamination. Outflux represents factors that decrease the system parameter, such as bacterial death or isolation.

Describe (in 2-3 sentences) the difference between algebraic and differential equations, focusing on the typical knowns and unknowns in each.

Algebraic equations typically involve finding a specific value of a variable given a known function. Differential equations, on the other hand, describe the relationship between an unknown function and its derivatives, often used when the specific function is not known directly.

Outline the steps to solve a separable differential equation of the form $\frac{dy}{dx} = f(x)g(y)$.

First, separate the variables to get $\frac{dy}{g(y)} = f(x)dx$. Second, integrate both sides, remembering the integration constant. Finally, rearrange the equation to obtain an explicit expression for y in terms of x and check your answer by differentiation.

Explain the concept of the 'initial condition' and its significance when solving differential equations.

An initial condition is a known value of the function at a specific point, often at $t = 0$. It is used to determine the arbitrary constant C in the general solution of a differential equation, leading to a unique solution that fits physical constraints. Signup and view all the answers

Why is exponential decay used to describe the activity of a drug in the body more accurately than a linear model?

Exponential decay is more accurate because the rate of decay is proportional to the amount of the drug present, unlike a linear model where the decay rate is constant. This reflects how the body processes drugs at a rate dependent on the substance's concentration. Signup and view all the answers

Explain in your own words how scientists can use differential equations to estimate the age of fossils.

Radioactive elements trapped in once-living materials decay exponentially at a known rate. By measuring the current amount of the radioactive element versus the initial amount and using differential equations, scientists can estimate backwards how old the material is, using the decay rate as a 'clock'. Signup and view all the answers

Explain the concept behind validating solutions to differential equations.

Validating solutions to DEs involves checking whether the solution satisfies the original differential equation by substituting it back into the equation. If the equation holds true, the solution is correct. Signup and view all the answers

In the context of enzyme kinetics, what is the significance of reaction catalysts and how do they influence reaction rates?

Reaction catalysts, such as enzymes, speed up reactions without being consumed in the process. They lower the activation energy required for reactions, significantly increasing the rate at which reactants are converted into products. Signup and view all the answers

Briefly describe the components of the enzymatic reaction equation: $E + S \rightleftharpoons ES \rightarrow E + P$.

E is the enzyme, S is the substrate, ES is the enzyme-substrate complex, and P is the product. The enzyme and substrate reversibly bind to form the enzyme-substrate complex, which then irreversibly transforms into the enzyme and the product. Signup and view all the answers

How is the Michaelis-Menten equation used to describe enzymatic reactions, and what parameters does it relate?

It is used to describe the rate of enzymatic reactions by relating the initial reaction rate ($V_0$) to the substrate concentration ([S]), the maximum reaction rate ($V_{max}$), and the Michaelis constant ($K_M$). It describes the rate of product formation as a function of substrate concentration. Signup and view all the answers

What does the Michaelis constant ($K_M$) represent in enzyme kinetics, and how is it determined graphically?

$K_M$ represents the substrate concentration at which the reaction rate is half of $V_{max}$. Graphically, it is determined by finding the substrate concentration on the x-axis that corresponds to $V_{max}/2$ on the y-axis of a Michaelis-Menten plot. Signup and view all the answers

Explain what is meant by 'enzyme saturation' and how it impacts the reaction velocity ($V_0$) at high substrate concentrations [S].

Enzyme saturation occurs when the concentration of substrate is so high that all available enzyme active sites are occupied. At high substrate concentrations, adding more substrate does not significantly increase the already fast reaction velocity as there are no more enzymes available to bind the substrate, and $V_0$ plateaus. Signup and view all the answers

Describe the fundamental difference between competitive enzyme inhibition and uncompetitive enzyme inhibition.

In competitive inhibition, the inhibitor binds to the active site on the enzyme and blocks it preventing the substrate from binding. In uncompetitive inhibition, the inhibitor binds only to the enzyme-substrate complex, preventing the complex from releasing product. Signup and view all the answers

How does the presence of a competitive inhibitor affect the $K_M$ and $V_{max}$ of an enzymatic reaction?

A competitive inhibitor increases the apparent $K_M$, reflecting a lower affinity for the substrate, but does not change the $V_{max}$ because at high substrate concentrations, the substrate can still outcompete the inhibitor. Signup and view all the answers

How does uncompetitive inhibition affect the $K_M$ and $V_{max}$ of an enzymatic reaction?

Uncompetitive inhibition decreases both the $K_M$ and $V_{max}$. The inhibitor binds to the enzyme substrate complex in such a way that less product is formed thus lowering the maximal rate, effectively reducing substrate affinity as well. Signup and view all the answers

Describe the effect of 'product inhibition' on an enzymatic reaction.

Product inhibition is where the product of a reaction acts as an inhibitor, typically a competitive inhibitor. As the product concentration increases less enzyme is avaliable to produce product slowing the entire reaction. Signup and view all the answers

Describe Robert Brown's observations, and explain its relationship to the concept of atoms.

Robert Brown saw jittery motion of pollen (and dust) in solution. Because this observation involved lifless particles it suggested that this motion was because of the sum of the impacts by tiny atoms or molecules. Signup and view all the answers

What key parameter does the diffusion equation relate, and what does it represent physically?

The diffusion equation relates the concetration of molecules as a function of time and space and the diffusion constant. Physically, it represents how the concetration of molecules change depending on the diffusion constant. Signup and view all the answers

Briefly describe how the Stokes-Einstein equation relates the diffusion constant to properties of the fluid and the diffusing particles.

The Stokes-Einstein equation relates the diffusion constant to the temperature, viscosity of the fluid, and the radius of the diffusing particles. Specifically, diffusion is proportional to the temperature but inversley proportional to the viscosity and radius. Signup and view all the answers

What does the drag coefficient describe in the context of a particle moving in a fluid, and how it is related to the particle's velocity and the drag force?

The drag coefficient quantifies how much resistance a particle experiences as it moves through a fluid. It determines the relationship between the force acting on the partiucle and its velocity. Signup and view all the answers

Describe the relationship between the distance a particle diffuses and time, according to the diffusion equation.

The distance a particle diffuses is proportional to the square root of time such that diffusion becomes significantly more ineffective as displacement increases. Signup and view all the answers

Why can larger organisms not rely on diffusion alone to transport substances throughout their body?

Diffusion is slow over long distances. Larger organisms need to make use of molecular tools and directed motion to move molecules around efficiently. Signup and view all the answers

Algebraic equations are often used in nanobiology. Give an example algebraic calculation one might perform on cells.

One might count the number of cells $N$ under a microscope, then assume each cell has a volume $V$, then the total mass of cells is $NVp$ where $p$ is the density of the cell. Signup and view all the answers

Bacteria need to swim at high speeds to avoid being consumed. They need to speed up from zero to speed $v$ in $t$ seconds. Write a differential equation that describe the relationship between position as a function of time $x(t)$.

Newtons's equation gives $F=ma$ where a is acceleration which is $d^2x/dt^2$. Thus $F = m d^2x/dt^2$. Finally $F$ is equal to the force from their motor minus the drag: $F= F_{motor} -6\pi \eta r dx/dt$. Signup and view all the answers

Carbon dating equations rely on the natural rate of radioactive decay. Does the amount that remains depend on the starting amount?

No it does not. If the half life is some amount $x$ for $N=600$, then it must also be $x$ when $N=300$. Signup and view all the answers

$K_M$ can be measured in a lab with tools like spectrophotometers. What aspect needs to be measured in order to calculate the value? Explain why one cannot choose $V_{max}$ instead.

One needs to measure the reaction rate $V_0$ for different values of the substrate. $V_{max}$ may depend on the amount of enzyme in the mixture. So by looking at the concetration where you get halfe of $V_{max}$, one can measure $K_M$. Signup and view all the answers

What value dictates the relative ammount of competitive binding and what values do these depend on?

The amount of competitive binding depends on the number of sites available. This in turn depends on the kinetic rate, how often the enzyme interacts with the substrate, and the concentrations of materials. Signup and view all the answers

Researchers find out there is contaminent in their enzyme sample that prevents their reaction from ever achieving $V_{max}$. What kind of binding might explain this?

Uncompetitive inhibitors prevent the complex from disassociating and thus a lower overall reaction velocity. The reaction will therefor not be able to acheive maximal levels. . Signup and view all the answers

Why do the very small insects called no-see-ums tend to stay around the coast?

These particular insects are very sensitive to air pressure needed to get oxygen through their system. Their size is dictated by how much diffusion their body can handle given the pressure. Sea-levels are simply better in terms of getting oxygen. Signup and view all the answers

Why do we see diffusion in areas like finance?

Money tends to move between different areas in the economy. Models for this also work mathematically as a diffusion model due to how the random walk is modeled. Signup and view all the answers

Name three processes that can be modeled using diffusion.

Molecules in solution, heat, flexible polymers, genetic drift, motile bacteria, and financial instruments can all be modeled through diffusion. Signup and view all the answers

Researchers are able to measure the drag force on a microsphere. Give one physical insight this number would help unlock.

It gives insight into the speed of movement but also an indirect glimpse into the diffusion coefficient as the parameters of the equation link the concepts. Thus researchers do not always need to directly measure these values. Signup and view all the answers

A researcher in the lab is growing a culture of bacteria. However, their incubator breaks and the temperature of the sample sharply drops! What will eventually happen to the bacteria?

Even if the bacteria do not immediately die, their movement and thus chance for resource accumulation is driven by diffusion. As temperature decreases so too will the bacteria's capacity for growth. Signup and view all the answers

A researcher is trying to reduce the noise in their experiment. They are unable to stop the table from shaking! How will they reduce the impact on their results?

The degree to which a particle diffuses will depend on its radius. They should use the biggest particles possible to measure as this would decrease the impact of any vibration. Signup and view all the answers

Scientists can inject spheres into the human body and observe their movement to learn about disease in the body. Name one obstacle these scientists need to overcome with their analysis.

Diffusion is only effective at short distances! So the sphere will take too long to mix. Solutions involve active transport or convection to move the spheres between areas! Signup and view all the answers

Flashcards

What is Nanobiology?

Study of biology at the molecular level, using physics and maths to describe biology.

What is a differential equation?

An equation that describes the relationship between an unknown function, its derivative(s), and the variable of the function.