Solutions to CZ1103 Final Exam 1998

  1. A man on a parachute jumps from an airplane. The position of the man is denoted by the height h above the ground, and the velocity is denoted as v (the velocity is negative, as it is pointing downward). The equations of motion can be written as

    2. dh/dt = v
    dv/dt = -g + bv2

    Here g and b are constants. What is the terminal velocity of the parachute? Write down a numerical scheme for solving these equations using Euler's method.

    Ans:

      

    dh/dt = v
dv/dt = -g + bv2
    Terminal Velocity is reached when dv/dt = 0 i.e. velocity no longer increases, and the velocity settles to a large fixed value. 
    dv/dt = 0
-g + bv2 = 0
v2 = g/b
v = +/- (g/b)0.5
therefore, terminal velocity = -(g/b)0.5
    Euler's Method: 
    vn+1 = vn + an dt
xn+1 = xn + vn dt

where a = F(x,v)/m

therefore:
vn+1 = vn + (-g + bvn2)dt
hn+1 = hn + vn dt
  2. Describe the main difference between fractals and simple geometric objects such as spheres and cylinders. Give an example of a computer-generated fractal object by describing the algorithm used to create it (you don't need to write MatLab codes; you only need to describe the algorithm used to generate it).

    3. Ans:

     Fractals are rough or fragmented geometric shapes which can be sub-divided into parts, each of which is at least approximately a reduced size copy of the whole.

    Simple geometric objects are those with regular or ordered shapes.

    An example algorithm for generating fractals: 

            Iterated Function Systems:  x -> Mx + c
        where M is a NxN matrix
              c is a N-component column vector
  3. To simulate global weather patterns, we typically need to discretize spatially the differential equations for various fields (such as pressure, temperature, velocity, etc). What are the main advantages and disadvantages of using a grid with a large grid spacing? Give a reason why long term weather forecasting might be difficult.

    4. Ans:

     Advantages:
    Introduction of grids with large spacing saves CPU time and memory requirements, which is particularly important for solving large systems.

    Disadvantages:
    Details of weather patterns cannot be described. Anything between 2 grid points is lost.

    Long term weather forecasting is difficult because the memory and CPU time requirement is beyond the capability of current computer systems. 


  4. Consider a molecular dynamics simulation of N particles in 3 dimensional space. The simulation is based on solving coupled first order ordinary differential equations (ODE) numerically. How many ODEs are needed? To obtain the solutions, what initial conditions are needed?

    5. Ans:

     For each particle in the system, Newton's 2nd Law is applied in each dimension: 

      Fx = max = m d2x/dt2
  Fy = may = m d2y/dt2
  Fz = maz = m d2z/dt2
        
  where x, y, z denote the position of the particle;
    vx, vy, vz denote the velocity of the particle ( vx = dx/dt, etc... )
    ax, ay, az denote the acceleration of the particle ( ax = dvx/dt = d2x/dt2, etc... )
    Therefore 3 coupled ODEs are needed for each particle. For N particles, the total number of ODEs is 3N.

    The initial conditions required:

    1. Starting position of each particle (x0, y0, z0)
    2. Starting velocity of each particle (Vx0, Vy0, Vz0)


  5. Consider a radioactive decay problem involving two types of nuclei, A and B, with population NA(t) and NB(t). Suppose that nuclei of type A decay into ones of type B, while nuclei of type B decay into ones of type A. Assume that the probability per unit time (denoted as k) is the same. Write down the rate equations for NA and NB (coupled differential equations for NA and NB). If A and B can react to give rise to a third type of nuclei C, write down again the rate equations.

    6. Ans:

     We model the system Using the simple population growth model:

    Nuclei A population: NA
    Nuclei B population: NB

    Population change (rate): dNA/dt and dNB/dt

    Relationship:

    Rate Equations: 
      dNA/dt = -kNA + kNB
  dNB/dt = -kNB + kNA
    A and B react to give rise to C, so: 
      NA decreases by kNANB
  NB decreases by kNANB
  NC increases by kNANB
    Therefore, Rate Equations: 
      dNA/dt = -kNA + kNB -kNANB
  dNB/dt = -kNB + kNA -kNANB
  dNC/dt = kNANB
  6. Name three graphic models used to represent the 3D structure of a molecule on computer. Describe the steps to view a 3D structure of a chemical compound you generated using the software ISIS draw. How to save the xyz coordinates of this compound?

    7. Wireframe, ball-and-stick, spacefill.
    To view: Use Chemistry => View Molecule in Rasmol to view the
            molecule in 3D.
    To save: Use File => Import => Molfile 


  7. In computer modeling of molecular mechanics of molecules, empirical energy functions are often used to describe atomic motions. Please write down the energy function for: (1) angle bending, (2) electrostatic interaction, (3) torsion, and (4) van der Waals interaction.

    8. (1) 

    (4) and (2) 

    (3) 
     
     

  8. Describe the purpose of the following databases:
    1. Brookhaven protein databank: 3D crystal and NMR structure

      2.     of proteins, DNA, RNA and ligand-bound complexes.
    2. SWISS-PROT:Annotated protein sequence database.
    3. SCOP: Structural classification of proteins.
    4. The national center for biotechnology information: Integrated ENTREZ retrieval software, BLAST search tools and databases for genetics, gene and protein sequences, 3D structures, and on-line PubMed library.
    5. The genome database: Datebase for genes of human and other species.



  9. Which is the best method to do computer-aided drug design when the 3D structure of the target protein is known? (1) Homology modeling, (2) Quantitative structure-activity relationship, and (3) Ligand-protein docking.

    Ligand-protein docking